a-number
stringlengths 7
7
| sequence
sequencelengths 1
377
| description
stringlengths 3
852
|
|---|---|---|
A000701 | [
"0",
"0",
"1",
"1",
"2",
"3",
"5",
"7",
"10",
"14",
"20",
"27",
"37",
"49",
"66",
"86",
"113",
"146",
"190",
"242",
"310",
"392",
"497",
"623",
"782",
"973",
"1212",
"1498",
"1851",
"2274",
"2793",
"3411",
"4163",
"5059",
"6142",
"7427",
"8972",
"10801",
"12989",
"15572",
"18646",
"22267",
"26561",
"31602",
"37556",
"44533",
"52743",
"62338",
"73593"
] | One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes. |
A000702 | [
"1",
"1",
"3",
"4",
"5",
"7",
"9",
"14",
"18",
"24",
"31",
"43",
"55",
"72",
"94",
"123",
"156",
"200",
"254",
"324",
"408",
"513",
"641",
"804",
"997",
"1236",
"1526",
"1883",
"2308",
"2829",
"3451",
"4209",
"5109",
"6194",
"7485",
"9038",
"10871",
"13063",
"15654",
"18738",
"22365",
"26665",
"31716",
"37682",
"44669",
"52887",
"62494",
"73767"
] | a(n) is the number of conjugacy classes in the alternating group A_n. |
A000703 | [
"4",
"6",
"7",
"7",
"8",
"9",
"9",
"10",
"10",
"10",
"11",
"11",
"12",
"12",
"12",
"13",
"13",
"13",
"13",
"14",
"14",
"14",
"15",
"15",
"15",
"15",
"16",
"16",
"16",
"16",
"16",
"17",
"17",
"17",
"17",
"18",
"18",
"18",
"18",
"18",
"19",
"19",
"19",
"19",
"19",
"19",
"20",
"20",
"20",
"20",
"20",
"21",
"21",
"21",
"21",
"21",
"21",
"22",
"22",
"22",
"22",
"22",
"22",
"22",
"23",
"23",
"23",
"23",
"23",
"23",
"24",
"24",
"24",
"24"
] | Chromatic number (or Heawood number) of nonorientable surface with n crosscaps. |
A000704 | [
"1",
"1",
"1",
"1",
"4",
"16",
"46",
"106",
"316",
"1324",
"5356",
"18316",
"63856",
"272416",
"1264264",
"5409496",
"22302736",
"101343376",
"507711376",
"2495918224",
"11798364736",
"58074029056",
"309240315616",
"1670570920096",
"8792390355904",
"46886941456576",
"264381946998976",
"1533013006902976"
] | Number of degree-n even permutations of order dividing 2. |
A000705 | [
"2",
"3",
"2",
"5",
"2",
"3",
"7",
"2",
"11",
"13",
"2",
"3",
"5",
"17",
"19",
"2",
"23",
"7",
"29",
"3",
"31",
"2",
"37",
"41",
"43",
"47",
"5",
"53",
"59",
"2",
"11",
"61",
"3",
"67",
"71",
"73",
"79",
"13",
"83",
"89",
"2",
"97",
"101",
"103",
"107",
"7",
"109",
"113",
"17",
"127",
"131",
"137",
"139",
"3",
"5",
"149",
"151",
"19",
"2",
"157",
"163",
"167",
"173",
"179",
"181",
"191",
"193",
"197",
"199"
] | n-th superior highly composite number A002201(n) is product of first n terms of this sequence. |
A000706 | [
"1",
"504",
"270648",
"144912096",
"77599626552",
"41553943041744",
"22251789971649504",
"11915647845248387520",
"6380729991419236488504",
"3416827666558895485479576",
"1829682703808504464920468048",
"979779820147442370107345764512"
] | Expansion of modular function 1/E_3 (cf. A013973). |
A000707 | [
"1",
"1",
"2",
"6",
"20",
"71",
"259",
"961",
"3606",
"13640",
"51909",
"198497",
"762007",
"2934764",
"11333950",
"43874857",
"170193528",
"661386105",
"2574320659",
"10034398370",
"39163212165",
"153027659730",
"598577118991",
"2343628878849",
"9184197395425",
"36020235035016",
"141376666307608"
] | Number of permutations of [1,2,...,n] with n-1 inversions. |
A000708 | [
"-1",
"-1",
"0",
"1",
"6",
"29",
"150",
"841",
"5166",
"34649",
"252750",
"1995181",
"16962726",
"154624469",
"1505035350",
"15583997521",
"171082318686",
"1985148989489",
"24279125761950",
"312193418011861",
"4210755676649046",
"59445878286889709",
"876726137720576550",
"13483686390543382201"
] | a(n) = E(n+1) - 2*E(n), where E(i) is the Euler number A000111(i). |
A000709 | [
"1",
"2",
"4",
"7",
"12",
"21",
"38",
"68",
"124",
"229",
"428",
"806",
"1530",
"2919",
"5591",
"10750",
"20730",
"40077",
"77653",
"150752",
"293161",
"570963",
"1113524",
"2174315",
"4250367",
"8317036",
"16289636",
"31931697",
"62642861",
"122980015",
"241595101",
"474910732",
"934088141",
"1838227618",
"3619356631"
] | Related to population of numbers of form x^2 + y^2. |
A000710 | [
"1",
"2",
"5",
"10",
"20",
"35",
"62",
"102",
"167",
"262",
"407",
"614",
"919",
"1345",
"1952",
"2788",
"3950",
"5524",
"7671",
"10540",
"14388",
"19470",
"26190",
"34968",
"46439",
"61275",
"80455",
"105047",
"136541",
"176593",
"227460",
"291673",
"372605",
"474085",
"601105",
"759380",
"956249",
"1200143",
"1501749",
"1873407"
] | Number of partitions of n, with two kinds of 1, 2, 3 and 4. |
A000711 | [
"1",
"3",
"9",
"22",
"51",
"107",
"217",
"416",
"775",
"1393",
"2446",
"4185",
"7028",
"11569",
"18749",
"29908",
"47083",
"73157",
"112396",
"170783",
"256972",
"383003",
"565961",
"829410",
"1206282",
"1741592",
"2497425",
"3557957",
"5037936",
"7091711",
"9927583",
"13823626",
"19151731",
"26404879",
"36236988",
"49509149"
] | Number of partitions of n, with three kinds of 1,2,3 and 4 and two kinds of 5,6,7,... |
A000712 | [
"1",
"2",
"5",
"10",
"20",
"36",
"65",
"110",
"185",
"300",
"481",
"752",
"1165",
"1770",
"2665",
"3956",
"5822",
"8470",
"12230",
"17490",
"24842",
"35002",
"49010",
"68150",
"94235",
"129512",
"177087",
"240840",
"326015",
"439190",
"589128",
"786814",
"1046705",
"1386930",
"1831065",
"2408658",
"3157789",
"4126070",
"5374390"
] | Generating function = Product_{m>=1} 1/(1 - x^m)^2; a(n) = number of partitions of n into parts of 2 kinds. |
A000713 | [
"1",
"3",
"8",
"18",
"38",
"74",
"139",
"249",
"434",
"734",
"1215",
"1967",
"3132",
"4902",
"7567",
"11523",
"17345",
"25815",
"38045",
"55535",
"80377",
"115379",
"164389",
"232539",
"326774",
"456286",
"633373",
"874213",
"1200228",
"1639418",
"2228546",
"3015360",
"4062065",
"5448995",
"7280060",
"9688718",
"12846507",
"16972577"
] | EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ... |
A000714 | [
"1",
"3",
"9",
"21",
"47",
"95",
"186",
"344",
"620",
"1078",
"1835",
"3045",
"4967",
"7947",
"12534",
"19470",
"29879",
"45285",
"67924",
"100820",
"148301",
"216199",
"312690",
"448738",
"639464",
"905024",
"1272837",
"1779237",
"2473065",
"3418655",
"4701611",
"6434015",
"8763676"
] | Number of partitions of n, with three kinds of 1 and 2 and two kinds of 3,4,5,.... |
A000715 | [
"1",
"3",
"9",
"22",
"50",
"104",
"208",
"394",
"724",
"1286",
"2229",
"3769",
"6253",
"10176",
"16303",
"25723",
"40055",
"61588",
"93647",
"140875",
"209889",
"309846",
"453565",
"658627",
"949310",
"1358589",
"1931464",
"2728547",
"3831654",
"5350119",
"7430158",
"10265669",
"14113795",
"19313168",
"26309405",
"35685523"
] | Number of partitions of n, with three kinds of 1,2 and 3 and two kinds of 4,5,6,.... |
A000716 | [
"1",
"3",
"9",
"22",
"51",
"108",
"221",
"429",
"810",
"1479",
"2640",
"4599",
"7868",
"13209",
"21843",
"35581",
"57222",
"90882",
"142769",
"221910",
"341649",
"521196",
"788460",
"1183221",
"1762462",
"2606604",
"3829437",
"5590110",
"8111346",
"11701998",
"16790136",
"23964594",
"34034391",
"48104069",
"67679109",
"94800537",
"132230021",
"183686994",
"254170332"
] | Number of partitions of n into parts of 3 kinds. |
A000717 | [
"1",
"1",
"1",
"3",
"6",
"24",
"148",
"1646",
"34040",
"1358852",
"106321628",
"16006173014",
"4525920859198",
"2404130854745735",
"2426376196165902704",
"4648429222263945620900",
"16788801124652327714275292",
"114722035311851620271616102401"
] | Number of graphs with n nodes and floor(n(n-1)/4) edges. |
A000718 | [
"1",
"2",
"6",
"20",
"65",
"226",
"883",
"3947",
"20089",
"115036",
"732171",
"5126901",
"39165917",
"324138010",
"2888934623",
"27587288507",
"281001801969",
"3041152133848",
"34849036364659",
"421526126267265",
"5367037330561365",
"71752003756908550"
] | Boustrophedon transform of triangular numbers 1,1,3,6,10,... |
A000719 | [
"0",
"0",
"1",
"2",
"5",
"13",
"44",
"191",
"1229",
"13588",
"288597",
"12297299",
"1031342116",
"166123498733",
"50668194387427",
"29104827043066808",
"31455591302381718651",
"64032471448906164191208",
"245999896712611657677614268",
"1787823725136869060356731751124"
] | Number of disconnected graphs with n nodes. |
A000720 | [
"0",
"1",
"2",
"2",
"3",
"3",
"4",
"4",
"4",
"4",
"5",
"5",
"6",
"6",
"6",
"6",
"7",
"7",
"8",
"8",
"8",
"8",
"9",
"9",
"9",
"9",
"9",
"9",
"10",
"10",
"11",
"11",
"11",
"11",
"11",
"11",
"12",
"12",
"12",
"12",
"13",
"13",
"14",
"14",
"14",
"14",
"15",
"15",
"15",
"15",
"15",
"15",
"16",
"16",
"16",
"16",
"16",
"16",
"17",
"17",
"18",
"18",
"18",
"18",
"18",
"18",
"19",
"19",
"19",
"19",
"20",
"20",
"21",
"21",
"21",
"21",
"21",
"21"
] | pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159... |
A000721 | [
"1",
"2",
"6",
"74",
"169112",
"39785643746726",
"37126652766640082937217814348006",
"558874591495497577231218517843968898077072559983411918227348931497772"
] | Number of balanced Boolean functions of n variables. |
A000722 | [
"1",
"2",
"24",
"40320",
"20922789888000",
"263130836933693530167218012160000000",
"126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000"
] | Number of invertible Boolean functions of n variables: a(n) = (2^n)!. |
A000723 | [
"1",
"3",
"840",
"54486432000",
"68523655451482690147713024000000",
"2753622660283944533494648206058191857701074569760095316814277221684346880000000000000"
] | Invertible Boolean functions of n variables. |
A000724 | [
"1",
"3",
"196",
"3406687200",
"2141364232858913975435172249600",
"43025354066936633335853878219659247776604712057098163541301459387254457761792000000"
] | Invertible Boolean functions of n variables. |
A000725 | [
"1",
"2",
"154",
"2270394624",
"571030462095782973206774552784",
"3824475917061034074298122508414160251634847335755905881951011420229530501911521280"
] | Invertible Boolean functions of n variables. |
A000726 | [
"1",
"1",
"2",
"2",
"4",
"5",
"7",
"9",
"13",
"16",
"22",
"27",
"36",
"44",
"57",
"70",
"89",
"108",
"135",
"163",
"202",
"243",
"297",
"355",
"431",
"513",
"617",
"731",
"874",
"1031",
"1225",
"1439",
"1701",
"1991",
"2341",
"2731",
"3197",
"3717",
"4333",
"5022",
"5834",
"6741",
"7803",
"8991",
"10375",
"11923",
"13716",
"15723",
"18038",
"20628",
"23603"
] | Number of partitions of n in which no parts are multiples of 3. |
A000727 | [
"1",
"-4",
"2",
"8",
"-5",
"-4",
"-10",
"8",
"9",
"0",
"14",
"-16",
"-10",
"-4",
"0",
"-8",
"14",
"20",
"2",
"0",
"-11",
"20",
"-32",
"-16",
"0",
"-4",
"14",
"8",
"-9",
"20",
"26",
"0",
"2",
"-28",
"0",
"-16",
"16",
"-28",
"-22",
"0",
"14",
"16",
"0",
"40",
"0",
"-28",
"26",
"32",
"-17",
"0",
"-32",
"-16",
"-22",
"0",
"-10",
"32",
"-34",
"-8",
"14",
"0",
"45",
"-4",
"38",
"8",
"0",
"0",
"-34",
"-8",
"38",
"0",
"-22",
"-56",
"2",
"-28",
"0",
"0",
"-10",
"20",
"64",
"-40",
"-20",
"44"
] | Expansion of Product_{k >= 1} (1 - x^k)^4. |
A000728 | [
"1",
"-5",
"5",
"10",
"-15",
"-6",
"-5",
"25",
"15",
"-20",
"9",
"-45",
"-5",
"25",
"20",
"10",
"15",
"20",
"-50",
"-35",
"-30",
"55",
"-50",
"15",
"80",
"1",
"50",
"-35",
"-45",
"-15",
"5",
"-50",
"-25",
"-55",
"85",
"51",
"50",
"10",
"-40",
"65",
"10",
"-10",
"-115",
"50",
"-115",
"-100",
"85",
"80",
"-30",
"5",
"20",
"45",
"70",
"65",
"45",
"-55",
"-100"
] | Expansion of Product_{n>=1} (1-x^n)^5. |
A000729 | [
"1",
"-6",
"9",
"10",
"-30",
"0",
"11",
"42",
"0",
"-70",
"18",
"-54",
"49",
"90",
"0",
"-22",
"-60",
"0",
"-110",
"0",
"81",
"180",
"-78",
"0",
"130",
"-198",
"0",
"-182",
"-30",
"90",
"121",
"84",
"0",
"0",
"210",
"0",
"-252",
"-102",
"-270",
"170",
"0",
"0",
"-69",
"330",
"0",
"-38",
"420",
"0",
"-190",
"-390",
"0",
"-108",
"0",
"0",
"0",
"-300",
"99",
"442",
"210",
"0",
"418",
"-294",
"0",
"0",
"-510",
"378",
"-540",
"138",
"0"
] | Expansion of Product_{k >= 1} (1 - x^k)^6. |
A000730 | [
"1",
"-7",
"14",
"7",
"-49",
"21",
"35",
"41",
"-49",
"-133",
"98",
"-21",
"126",
"112",
"-176",
"-105",
"-126",
"140",
"-35",
"147",
"259",
"98",
"-420",
"-224",
"238",
"-455",
"273",
"-14",
"322",
"406",
"-35",
"-7",
"-637",
"-196",
"245",
"-181",
"-574",
"462",
"147",
"924",
"217",
"-329",
"-140",
"-7",
"-371",
"-777"
] | Expansion of Product_{n>=1} (1 - x^n)^7. |
A000731 | [
"1",
"-8",
"20",
"0",
"-70",
"64",
"56",
"0",
"-125",
"-160",
"308",
"0",
"110",
"0",
"-520",
"0",
"57",
"560",
"0",
"0",
"182",
"-512",
"-880",
"0",
"1190",
"-448",
"884",
"0",
"0",
"0",
"-1400",
"0",
"-1330",
"1000",
"1820",
"0",
"-646",
"1280",
"0",
"0",
"-1331",
"-2464",
"380",
"0",
"1120",
"0",
"2576",
"0",
"0",
"-880",
"1748",
"0",
"-3850",
"0",
"-3400",
"0",
"2703",
"4160",
"-2500",
"0",
"3458"
] | Expansion of Product (1 - x^k)^8 in powers of x. |
A000732 | [
"1",
"3",
"8",
"22",
"66",
"222",
"862",
"3838",
"19542",
"111894",
"712282",
"4987672",
"38102844",
"315339898",
"2810523166",
"26838510154",
"273374835624",
"2958608945772",
"33903161435148",
"410085034127000",
"5221364826476796",
"69804505809732988"
] | Boustrophedon transform of 1 & primes: 1,2,3,5,7,... |
A000733 | [
"1",
"2",
"4",
"10",
"30",
"101",
"394",
"1760",
"8970",
"51368",
"326991",
"2289669",
"17491625",
"144760655",
"1290204758",
"12320541392",
"125496010615",
"1358185050788",
"15563654383395",
"188254471337718",
"2396930376564860",
"32044598671291610"
] | Boustrophedon transform of partition numbers 1, 1, 1, 2, 3, 5, 7, ... |
A000734 | [
"1",
"2",
"5",
"15",
"49",
"177",
"715",
"3255",
"16689",
"95777",
"609875",
"4270695",
"32624329",
"269995377",
"2406363835",
"22979029335",
"234062319969",
"2533147494977",
"29027730898595",
"351112918079175",
"4470508510495609",
"59766296291090577"
] | Boustrophedon transform of 1,1,2,4,8,16,32,... |
A000735 | [
"1",
"-12",
"54",
"-88",
"-99",
"540",
"-418",
"-648",
"594",
"836",
"1056",
"-4104",
"-209",
"4104",
"-594",
"4256",
"-6480",
"-4752",
"-298",
"5016",
"17226",
"-12100",
"-5346",
"-1296",
"-9063",
"-7128",
"19494",
"29160",
"-10032",
"-7668",
"-34738",
"8712",
"-22572",
"21812",
"49248",
"-46872",
"67562",
"2508",
"-47520",
"-76912",
"-25191",
"67716"
] | Expansion of Product_{k>=1} (1 - x^k)^12. |
A000736 | [
"1",
"2",
"4",
"10",
"32",
"120",
"513",
"2455",
"13040",
"76440",
"492231",
"3465163",
"26530503",
"219754535",
"1959181266",
"18710532565",
"190588702776",
"2062664376064",
"23636408157551",
"285900639990875",
"3640199365715769",
"48665876423760247"
] | Boustrophedon transform of Catalan numbers 1, 1, 1, 2, 5, 14, ... |
A000737 | [
"1",
"3",
"8",
"21",
"60",
"197",
"756",
"3367",
"17136",
"98153",
"624804",
"4375283",
"33424512",
"276622829",
"2465449252",
"23543304919",
"239810132288",
"2595353815825",
"29740563986500",
"359735190398875",
"4580290700420064",
"61233976084442741"
] | Boustrophedon transform of natural numbers, cf. A000027. |
A000738 | [
"0",
"1",
"3",
"8",
"25",
"85",
"334",
"1497",
"7635",
"43738",
"278415",
"1949531",
"14893000",
"123254221",
"1098523231",
"10490117340",
"106851450165",
"1156403632189",
"13251409502982",
"160286076269309",
"2040825708462175",
"27283829950774822",
"382127363497453243",
"5595206208670390323"
] | Boustrophedon transform (first version) of Fibonacci numbers 0,1,1,2,3,... |
A000739 | [
"1",
"-16",
"104",
"-320",
"260",
"1248",
"-3712",
"1664",
"6890",
"-7280",
"-5568",
"-4160",
"33176",
"4640",
"-74240",
"29824",
"14035",
"54288",
"27040",
"-142720",
"1508",
"-110240",
"289536",
"222720",
"-380770",
"-83200",
"-123904",
"142912",
"7640",
"408000",
"386048"
] | Expansion of Product_{k>=1} (1 - x^k)^16. |
A000740 | [
"1",
"1",
"3",
"6",
"15",
"27",
"63",
"120",
"252",
"495",
"1023",
"2010",
"4095",
"8127",
"16365",
"32640",
"65535",
"130788",
"262143",
"523770",
"1048509",
"2096127",
"4194303",
"8386440",
"16777200",
"33550335",
"67108608",
"134209530",
"268435455",
"536854005",
"1073741823",
"2147450880"
] | Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement; also Dirichlet convolution of b_n=2^(n-1) with mu(n); also number of components of Mandelbrot set corresponding to Julia sets with an attractive n-cycle. |
A000741 | [
"0",
"0",
"1",
"3",
"6",
"9",
"15",
"18",
"27",
"30",
"45",
"42",
"66",
"63",
"84",
"84",
"120",
"99",
"153",
"132",
"174",
"165",
"231",
"180",
"270",
"234",
"297",
"270",
"378",
"276",
"435",
"360",
"450",
"408",
"540",
"414",
"630",
"513",
"636",
"552",
"780",
"558",
"861",
"690",
"828",
"759",
"1035",
"744",
"1113",
"870",
"1104",
"972",
"1326",
"945",
"1380",
"1116",
"1386",
"1218"
] | Number of compositions of n into 3 ordered relatively prime parts. |
A000742 | [
"1",
"4",
"10",
"20",
"34",
"56",
"80",
"120",
"154",
"220",
"266",
"360",
"420",
"560",
"614",
"816",
"884",
"1120",
"1210",
"1540",
"1572",
"2020",
"2080",
"2544",
"2638",
"3276",
"3200",
"4060",
"4040",
"4840",
"4896",
"5960",
"5710",
"7140",
"6954",
"8216",
"8136",
"9880",
"9244",
"11480",
"11010",
"12824",
"12650",
"15180",
"14024",
"17276"
] | Number of compositions of n into 4 ordered relatively prime parts. |
A000743 | [
"1",
"5",
"15",
"35",
"70",
"125",
"210",
"325",
"495",
"700",
"1000",
"1330",
"1820",
"2305",
"3060",
"3750",
"4830",
"5775",
"7315",
"8490",
"10625",
"12155",
"14880",
"16835",
"20475",
"22620",
"27405",
"30100",
"35750",
"39100",
"46360",
"49655",
"58905",
"62985",
"73320",
"78340",
"91390",
"95720",
"111930",
"117425"
] | Number of compositions of n into 5 ordered relatively prime parts. |
A000744 | [
"1",
"2",
"5",
"14",
"42",
"144",
"563",
"2526",
"12877",
"73778",
"469616",
"3288428",
"25121097",
"207902202",
"1852961189",
"17694468210",
"180234349762",
"1950592724756",
"22352145975707",
"270366543452702",
"3442413745494957",
"46021681757269830"
] | Boustrophedon transform (second version) of Fibonacci numbers 1,1,2,3,... |
A000745 | [
"1",
"5",
"18",
"57",
"180",
"617",
"2400",
"10717",
"54544",
"312353",
"1988104",
"13921501",
"106350816",
"880162337",
"7844596536",
"74910367309",
"763030711936",
"8257927397569",
"94628877364936",
"1144609672707741",
"14573622985067744",
"194834987492011649",
"2728787718495477144",
"39955604972310966797"
] | Boustrophedon transform of squares. |
A000746 | [
"1",
"4",
"13",
"39",
"120",
"407",
"1578",
"7042",
"35840",
"205253",
"1306454",
"9148392",
"69887664",
"578392583",
"5155022894",
"49226836114",
"501420422112",
"5426640606697",
"62184720675718",
"752172431553308",
"9576956842743904",
"128034481788227195"
] | Boustrophedon transform of triangular numbers. |
A000747 | [
"2",
"5",
"13",
"35",
"103",
"345",
"1325",
"5911",
"30067",
"172237",
"1096319",
"7677155",
"58648421",
"485377457",
"4326008691",
"41310343279",
"420783672791",
"4553946567241",
"52184383350787",
"631210595896453",
"8036822912123765",
"107444407853010597"
] | Boustrophedon transform of primes. |
A000748 | [
"1",
"-3",
"6",
"-9",
"9",
"0",
"-27",
"81",
"-162",
"243",
"-243",
"0",
"729",
"-2187",
"4374",
"-6561",
"6561",
"0",
"-19683",
"59049",
"-118098",
"177147",
"-177147",
"0",
"531441",
"-1594323",
"3188646",
"-4782969",
"4782969",
"0",
"-14348907",
"43046721",
"-86093442",
"129140163",
"-129140163",
"0",
"387420489",
"-1162261467"
] | Expansion of bracket function. |
A000749 | [
"0",
"0",
"0",
"1",
"4",
"10",
"20",
"36",
"64",
"120",
"240",
"496",
"1024",
"2080",
"4160",
"8256",
"16384",
"32640",
"65280",
"130816",
"262144",
"524800",
"1049600",
"2098176",
"4194304",
"8386560",
"16773120",
"33550336",
"67108864",
"134225920",
"268451840",
"536887296",
"1073741824",
"2147450880"
] | a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3), n > 3, with a(0)=a(1)=a(2)=0, a(3)=1. |
A000750 | [
"1",
"-5",
"15",
"-35",
"70",
"-125",
"200",
"-275",
"275",
"0",
"-1000",
"3625",
"-9500",
"21250",
"-42500",
"76875",
"-124375",
"171875",
"-171875",
"0",
"621875",
"-2250000",
"5890625",
"-13171875",
"26343750",
"-47656250",
"77109375",
"-106562500",
"106562500",
"0"
] | Expansion of bracket function. |
A000751 | [
"1",
"2",
"5",
"14",
"42",
"143",
"555",
"2485",
"12649",
"72463",
"461207",
"3229622",
"24671899",
"204185616",
"1819837153",
"17378165240",
"177012514388",
"1915724368181",
"21952583954117",
"265533531724484",
"3380877926676504",
"45199008472762756"
] | Boustrophedon transform of partition numbers. |
A000752 | [
"1",
"3",
"9",
"28",
"93",
"338",
"1369",
"6238",
"31993",
"183618",
"1169229",
"8187598",
"62545893",
"517622498",
"4613366689",
"44054301358",
"448733127793",
"4856429646978",
"55650582121749",
"673136951045518",
"8570645832753693",
"114581094529057058",
"1604780986816602409",
"23497612049668468078"
] | Boustrophedon transform of powers of 2. |
A000753 | [
"1",
"2",
"5",
"16",
"59",
"243",
"1101",
"5461",
"29619",
"175641",
"1137741",
"8031838",
"61569345",
"510230087",
"4549650423",
"43452408496",
"442620720531",
"4790322653809",
"54893121512453",
"663974736739232",
"8453986695437957",
"113021461431438475"
] | Boustrophedon transform of Catalan numbers. |
A000754 | [
"1",
"4",
"12",
"33",
"96",
"317",
"1218",
"5425",
"27608",
"158129",
"1006574",
"7048657",
"53847420",
"445643681",
"3971876930",
"37928628529",
"386337833232",
"4181155148673",
"47912508680086",
"579538956964241",
"7378919177090244",
"98648882783190305"
] | Boustrophedon transform of odd numbers. |
A000755 | [
"0",
"1",
"2",
"11",
"32",
"50",
"132",
"380",
"368",
"1135",
"1120",
"4348",
"3622",
"10568",
"30634",
"46304",
"55576",
"152210"
] | No-3-in-line problem on n X n grid: total number of ways of placing 2n points on n X n grid so no 3 are in a line. No symmetries are taken into account. |
A000756 | [
"1",
"2",
"3",
"5",
"13",
"41",
"157",
"699",
"3561",
"20401",
"129881",
"909523",
"6948269",
"57504201",
"512516565",
"4894172027",
"49851629137",
"539521049441",
"6182455849009",
"74781598946211",
"952148890494165",
"12729293006112121",
"178281831561868013"
] | Boustrophedon transform of sequence 1,1,0,0,0,0,... |
A000757 | [
"1",
"0",
"0",
"1",
"1",
"8",
"36",
"229",
"1625",
"13208",
"120288",
"1214673",
"13469897",
"162744944",
"2128047988",
"29943053061",
"451123462673",
"7245940789072",
"123604151490592",
"2231697509543361",
"42519034050101745",
"852495597142800376"
] | Number of cyclic permutations of [n] with no i->i+1 (mod n) |
A000758 | [
"0",
"1",
"5",
"20",
"76",
"285",
"1066",
"3991",
"14976",
"56353"
] | Related to cumulative height of rooted plane trees. |
A000759 | [
"1",
"4",
"12",
"44",
"172",
"772",
"3308",
"14924",
"64956",
"294252",
"1301044",
"5930588",
"26506948",
"121290940",
"546050988",
"2505533940",
"11340303508",
"52147596788",
"236995050900",
"1091701675948",
"4977541017540",
"22961416861940",
"104965762062612"
] | Number of n-step self-avoiding walks on cubic lattice ending at point with x=0. |
A000760 | [
"1",
"8",
"40",
"176",
"748",
"3248",
"14280",
"63768",
"285296",
"1285688",
"5794436",
"26261224",
"119028156",
"541876608",
"2466620624",
"11267536496",
"51458718144",
"235690960392",
"1079212461992",
"4953659984000",
"22730713367468",
"104520944666808"
] | Number of n-step self-avoiding walks on cubic lattice ending at point with x=1. |
A000761 | [
"1",
"12",
"84",
"468",
"2332",
"11068",
"51472",
"237832",
"1095384",
"5040568",
"23168528",
"106496816",
"489379904",
"2250000884",
"10345888480",
"47604198576",
"219096141188",
"1009071461380",
"4648802248764",
"21431064157200",
"98828123716260"
] | Number of n-step self-avoiding walks on cubic lattice ending at point with x=2. |
A000762 | [
"1",
"16",
"144",
"984",
"5756",
"30760",
"155912",
"766424",
"3698848",
"17648312",
"83558828",
"393534176",
"1846227984",
"8637479208",
"40325165648",
"187980582568",
"875268197452",
"4072021100336",
"18931821861960",
"87979249474568"
] | Number of n-step self-avoiding walks on cubic lattice ending at point with x=3. |
A000763 | [
"1",
"3",
"19",
"195",
"2831",
"53703",
"1264467",
"35661979",
"1173865927",
"44218244943",
"1877050837355",
"88693432799667",
"4618194424504623",
"262771389992099719",
"16223185411792992403",
"1080238361814167993739",
"77171781603974127429527"
] | Number of interval orders constructed from n intervals of generic lengths. |
A000764 | [
"1",
"2",
"5",
"16",
"60",
"258",
"1247",
"6686",
"39371",
"252688",
"1756920",
"13168178",
"105949517",
"911834394",
"8367625793",
"81642384468",
"844718036940",
"9245285569526",
"106790005796627",
"1298920385093126",
"16602066548692623"
] | Boustrophedon transform of Bell numbers. |
A000765 | [
"1",
"4",
"36",
"308",
"2764",
"25404",
"237164",
"2237948",
"21286548",
"203701772",
"1958748676",
"18908954324",
"183135542956",
"1778581016076",
"17314029758828",
"168891863875652"
] | Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 0. |
A000766 | [
"4",
"32",
"292",
"2672",
"24780",
"232512",
"2201948",
"20997008",
"201314448",
"1938659936",
"18737136032",
"181646110192",
"1765522809468",
"17198432462368",
"167859941774728"
] | Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 1. |
A000767 | [
"16",
"192",
"2016",
"20160",
"197940",
"1930944",
"18805488",
"183156320",
"1785303660",
"17421627280",
"170214459928",
"1665089608504",
"16307758577692",
"159896665015064"
] | Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 2. |
A000768 | [
"64",
"1024",
"12480",
"137472",
"1443616",
"14786176",
"149371964",
"1496777088",
"14924375156",
"148353336272",
"1471838989872",
"14584911842000",
"144423054478680"
] | Number of n-step self-avoiding walks on f.c.c. lattice ending at point with x = 3. |
A000769 | [
"0",
"1",
"1",
"4",
"5",
"11",
"22",
"57",
"51",
"156",
"158",
"566",
"499",
"1366",
"3978",
"5900",
"7094",
"19204"
] | No-3-in-line problem: number of inequivalent ways of placing 2n points on an n X n grid so that no 3 are in a line. |
A000770 | [
"1",
"21",
"266",
"2646",
"22827",
"179487",
"1323652",
"9321312",
"63436373",
"420693273",
"2734926558",
"17505749898",
"110687251039",
"693081601779",
"4306078895384",
"26585679462804",
"163305339345225",
"998969857983405",
"6090236036084530",
"37026417000002430",
"224595186974125331"
] | Stirling numbers of the second kind, S(n,6). |
A000771 | [
"1",
"28",
"462",
"5880",
"63987",
"627396",
"5715424",
"49329280",
"408741333",
"3281882604",
"25708104786",
"197462483400",
"1492924634839",
"11143554045652",
"82310957214948",
"602762379967440",
"4382641999117305",
"31677463851804540",
"227832482998716310",
"1631853797991016600"
] | Stirling numbers of second kind, S(n,7). |
A000772 | [
"1",
"1",
"2",
"6",
"23",
"107",
"583",
"3633",
"25444",
"197620",
"1684295",
"15618141",
"156453857",
"1683050189",
"19344093070",
"236497985706",
"3063827565763",
"41916787157011",
"603799270943519",
"9132945141812301",
"144708157060239704",
"2396568154933265024",
"41403636316192616995"
] | E.g.f. exp(tan(x) + sec(x) - 1). |
A000773 | [
"0",
"0",
"0",
"1",
"1",
"6",
"8",
"29",
"45",
"130",
"220",
"561",
"1001",
"2366",
"4368",
"9829",
"18565",
"40410",
"77540",
"164921",
"320001",
"669526",
"1309528",
"2707629",
"5326685",
"10919090",
"21572460",
"43942081",
"87087001",
"176565486",
"350739488",
"708653429",
"1410132405",
"2841788170",
"5662052980"
] | Number of numbers == 0 (mod 3) in range 2^n to 2^(n+1) with odd number of 1's in binary expansion. |
A000774 | [
"1",
"2",
"5",
"17",
"74",
"394",
"2484",
"18108",
"149904",
"1389456",
"14257440",
"160460640",
"1965444480",
"26029779840",
"370643938560",
"5646837369600",
"91657072281600",
"1579093018675200",
"28779361764249600",
"553210247226470400",
"11185850044938240000",
"237335752951879680000"
] | a(n) = n!*(1 + Sum_{i=1..n} 1/i). |
A000775 | [
"1",
"4",
"12",
"46",
"220",
"1268",
"8568",
"66456",
"582048",
"5681952",
"61174080",
"720089280",
"9199906560",
"126783809280",
"1874605662720",
"29601115891200",
"497155992883200",
"8849184886886400",
"166399076525875200",
"3296032301811916800",
"68596838245232640000",
"1496490349337948160000"
] | a(n) = n! * (n + 1 + 2*Sum_{k=1...n} 1/k). |
A000776 | [
"1",
"3",
"8",
"28",
"124",
"668",
"4248",
"31176",
"259488",
"2416032",
"24886080",
"281004480",
"3451887360",
"45832538880",
"654109585920",
"9986000371200",
"162391354675200",
"2802498609254400",
"51156349822771200",
"984775394044108800",
"19938798081699840000",
"423580563732049920000"
] | a(n) = n! * (1 + 2*Sum_{k=1..n} 1/k). |
A000777 | [
"1",
"2",
"7",
"24",
"83",
"293",
"1055",
"3860",
"14299",
"53481",
"201551",
"764217",
"2912167",
"11143499",
"42791039",
"164812364",
"636438059",
"2463251009",
"9552773999",
"37112526989",
"144410649239",
"562724141459",
"2195581527359",
"8576490341249",
"33537507830423",
"131272552839203",
"514285886020255"
] | a(n) = (n+2)*Catalan(n) - 1. |
A000778 | [
"1",
"2",
"6",
"18",
"55",
"173",
"560",
"1858",
"6291",
"21657",
"75581",
"266797",
"950911",
"3417339",
"12369284",
"45052514",
"165002459",
"607283489",
"2244901889",
"8331383609",
"31030387439",
"115948830659",
"434542177289",
"1632963760973",
"6151850548775",
"23229299473603",
"87900903988155"
] | a(n) = Catalan(n) + Catalan(n+1) - 1. |
A000779 | [
"1",
"4",
"22",
"162",
"1506",
"16950",
"224190",
"3408930",
"58596930",
"1123663590",
"23782729950",
"550718680050",
"13849716607650",
"375904338960150",
"10952237584237950",
"340947694234397250",
"11294123783425733250",
"396665528378000631750"
] | a(n) = 2*(2n-1)!!-(n-1)!*2^(n-1), where (2n-1)!! is A001147(n). |
A000780 | [
"1",
"4",
"16",
"78",
"456",
"3120",
"24480",
"216720",
"2136960",
"23224320",
"275788800",
"3552595200",
"49337164800",
"734788454400",
"11681891020800",
"197458829568000",
"3535951491072000",
"66869236482048000",
"1331693730791424000",
"27856727993622528000",
"610658404052336640000"
] | a(n) = (n+1)!/2 + (n-1)(n-1)!. |
A000781 | [
"1",
"4",
"12",
"36",
"111",
"353",
"1154",
"3860",
"13155",
"45525",
"159561",
"565249",
"2020687",
"7280419",
"26410094",
"96378164",
"353576699",
"1303271309",
"4824150869",
"17925098069",
"66834680639",
"249981423899",
"937696277309",
"3526652828321",
"13295935057031",
"50240112815003"
] | a(n) = 3*Catalan(n) - Catalan(n-1) - 1. |
A000782 | [
"1",
"3",
"8",
"23",
"70",
"222",
"726",
"2431",
"8294",
"28730",
"100776",
"357238",
"1277788",
"4605980",
"16715250",
"61020495",
"223931910",
"825632610",
"3056887680",
"11360977650",
"42368413620",
"158498860260",
"594636663660",
"2236748680998",
"8433988655580",
"31872759742852",
"120699748759856"
] | a(n) = 2*Catalan(n) - Catalan(n-1). |
A000783 | [
"4",
"341",
"91",
"15",
"124",
"35",
"25",
"9",
"28",
"33",
"15",
"65",
"21",
"15",
"341",
"51",
"45",
"25",
"45",
"21",
"55",
"69",
"33",
"25",
"28",
"27",
"65",
"87",
"35",
"49",
"49",
"33",
"85",
"35",
"51",
"91",
"45",
"39",
"95",
"91",
"105",
"205",
"77",
"45",
"76",
"133",
"65",
"49",
"66",
"51",
"65",
"85",
"65"
] | Erroneous version of A007535. |
A000784 | [
"0",
"1",
"2",
"2",
"4",
"6",
"6",
"11",
"16",
"20",
"28",
"41",
"51",
"70",
"93",
"122",
"158",
"211",
"266",
"350",
"450",
"577",
"730",
"948",
"1186",
"1510",
"1901",
"2408",
"2999",
"3790",
"4703",
"5898",
"7310",
"9111",
"11231",
"13979",
"17168",
"21229",
"26036",
"32095",
"39188",
"48155",
"58657",
"71798",
"87262",
"106472",
"129014"
] | Number of symmetrical planar partitions of n (planar partitions (A000219) that when regarded as 3-D objects have just one symmetry plane). |
A000785 | [
"0",
"0",
"0",
"1",
"2",
"5",
"11",
"21",
"39",
"73",
"129",
"226",
"388",
"659",
"1100",
"1821",
"2976",
"4828",
"7754",
"12370",
"19574",
"30789",
"48097",
"74725",
"115410",
"177366",
"271159",
"412665",
"625098",
"942932",
"1416362",
"2119282",
"3158840",
"4691431",
"6942882",
"10240503",
"15054705"
] | Number of asymmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have no symmetry. |
A000786 | [
"1",
"1",
"1",
"2",
"4",
"6",
"11",
"19",
"33",
"55",
"95",
"158",
"267",
"442",
"731",
"1193",
"1947",
"3137",
"5039",
"8026",
"12726",
"20024",
"31373",
"48835",
"75673",
"116606",
"178889",
"273061",
"415086",
"628115",
"946723",
"1421082",
"2125207",
"3166152",
"4700564",
"6954151",
"10254486",
"15071903"
] | Number of inequivalent planar partitions of n, when considering them as 3D objects. |
A000787 | [
"0",
"1",
"8",
"11",
"69",
"88",
"96",
"101",
"111",
"181",
"609",
"619",
"689",
"808",
"818",
"888",
"906",
"916",
"986",
"1001",
"1111",
"1691",
"1881",
"1961",
"6009",
"6119",
"6699",
"6889",
"6969",
"8008",
"8118",
"8698",
"8888",
"8968",
"9006",
"9116",
"9696",
"9886",
"9966",
"10001",
"10101",
"10801",
"11011",
"11111",
"11811",
"16091",
"16191"
] | Strobogrammatic numbers: the same upside down. |
A000788 | [
"0",
"1",
"2",
"4",
"5",
"7",
"9",
"12",
"13",
"15",
"17",
"20",
"22",
"25",
"28",
"32",
"33",
"35",
"37",
"40",
"42",
"45",
"48",
"52",
"54",
"57",
"60",
"64",
"67",
"71",
"75",
"80",
"81",
"83",
"85",
"88",
"90",
"93",
"96",
"100",
"102",
"105",
"108",
"112",
"115",
"119",
"123",
"128",
"130",
"133",
"136",
"140",
"143",
"147",
"151",
"156",
"159",
"163",
"167",
"172",
"176",
"181",
"186"
] | Total number of 1's in binary expansions of 0, ..., n. |
A000789 | [
"2",
"5",
"8",
"13",
"16",
"21",
"26",
"35",
"38",
"45",
"48"
] | Maximal order of a triangle-free cyclic graph with no independent set of size n. |
A000790 | [
"4",
"4",
"341",
"6",
"4",
"4",
"6",
"6",
"4",
"4",
"6",
"10",
"4",
"4",
"14",
"6",
"4",
"4",
"6",
"6",
"4",
"4",
"6",
"22",
"4",
"4",
"9",
"6",
"4",
"4",
"6",
"6",
"4",
"4",
"6",
"9",
"4",
"4",
"38",
"6",
"4",
"4",
"6",
"6",
"4",
"4",
"6",
"46",
"4",
"4",
"10",
"6",
"4",
"4",
"6",
"6",
"4",
"4",
"6",
"15",
"4",
"4",
"9",
"6",
"4",
"4",
"6",
"6",
"4",
"4",
"6",
"9",
"4",
"4",
"15",
"6",
"4",
"4",
"6",
"6",
"4",
"4",
"6",
"21",
"4",
"4",
"10",
"6",
"4"
] | Primary pretenders: least composite c such that n^c == n (mod c). |
A000791 | [
"3",
"6",
"9",
"14",
"18",
"23",
"28",
"36"
] | Ramsey numbers R(3,n). |
A000792 | [
"1",
"1",
"2",
"3",
"4",
"6",
"9",
"12",
"18",
"27",
"36",
"54",
"81",
"108",
"162",
"243",
"324",
"486",
"729",
"972",
"1458",
"2187",
"2916",
"4374",
"6561",
"8748",
"13122",
"19683",
"26244",
"39366",
"59049",
"78732",
"118098",
"177147",
"236196",
"354294",
"531441",
"708588",
"1062882",
"1594323",
"2125764",
"3188646",
"4782969",
"6377292"
] | a(n) = max{(n - i)*a(i) : i < n}; a(0) = 1. |
A000793 | [
"1",
"1",
"2",
"3",
"4",
"6",
"6",
"12",
"15",
"20",
"30",
"30",
"60",
"60",
"84",
"105",
"140",
"210",
"210",
"420",
"420",
"420",
"420",
"840",
"840",
"1260",
"1260",
"1540",
"2310",
"2520",
"4620",
"4620",
"5460",
"5460",
"9240",
"9240",
"13860",
"13860",
"16380",
"16380",
"27720",
"30030",
"32760",
"60060",
"60060",
"60060",
"60060",
"120120"
] | Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n. |
A000794 | [
"1",
"2",
"24",
"3852",
"18534400",
"4598378639550"
] | Permanent of projective plane of order n. |
A000795 | [
"1",
"2",
"12",
"152",
"3472",
"126752",
"6781632",
"500231552",
"48656756992",
"6034272215552",
"929327412759552",
"174008703107274752",
"38928735228629389312",
"10255194381004799025152",
"3142142941901073853366272",
"1107912434323301224813002752",
"445427836895850552387642130432"
] | Salié numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!. |
A000796 | [
"3",
"1",
"4",
"1",
"5",
"9",
"2",
"6",
"5",
"3",
"5",
"8",
"9",
"7",
"9",
"3",
"2",
"3",
"8",
"4",
"6",
"2",
"6",
"4",
"3",
"3",
"8",
"3",
"2",
"7",
"9",
"5",
"0",
"2",
"8",
"8",
"4",
"1",
"9",
"7",
"1",
"6",
"9",
"3",
"9",
"9",
"3",
"7",
"5",
"1",
"0",
"5",
"8",
"2",
"0",
"9",
"7",
"4",
"9",
"4",
"4",
"5",
"9",
"2",
"3",
"0",
"7",
"8",
"1",
"6",
"4",
"0",
"6",
"2",
"8",
"6",
"2",
"0",
"8",
"9",
"9",
"8",
"6",
"2",
"8",
"0",
"3",
"4",
"8",
"2",
"5",
"3",
"4",
"2",
"1",
"1",
"7",
"0",
"6",
"7",
"9",
"8",
"2",
"1",
"4"
] | Decimal expansion of Pi (or digits of Pi). |
A000797 | [
"17",
"27",
"33",
"52",
"73",
"82",
"83",
"103",
"107",
"137",
"153",
"162",
"217",
"219",
"227",
"237",
"247",
"258",
"268",
"271",
"282",
"283",
"302",
"303",
"313",
"358",
"383",
"432",
"437",
"443",
"447",
"502",
"548",
"557",
"558",
"647",
"662",
"667",
"709",
"713",
"718",
"722",
"842",
"863",
"898",
"953",
"1007",
"1117",
"1118"
] | Numbers that are not the sum of 4 tetrahedral numbers. |
A000798 | [
"1",
"1",
"4",
"29",
"355",
"6942",
"209527",
"9535241",
"642779354",
"63260289423",
"8977053873043",
"1816846038736192",
"519355571065774021",
"207881393656668953041",
"115617051977054267807460",
"88736269118586244492485121",
"93411113411710039565210494095",
"134137950093337880672321868725846",
"261492535743634374805066126901117203"
] | Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements. |
A000799 | [
"2",
"2",
"2",
"4",
"6",
"10",
"18",
"32",
"56",
"102",
"186",
"341",
"630",
"1170",
"2184",
"4096",
"7710",
"14563",
"27594",
"52428",
"99864",
"190650",
"364722",
"699050",
"1342177",
"2581110",
"4971026",
"9586980",
"18512790",
"35791394",
"69273666",
"134217728",
"260301048",
"505290270",
"981706810",
"1908874353"
] | a(n) = floor(2^n / n). |
A000800 | [
"1",
"1",
"1",
"2",
"5",
"13",
"38",
"125",
"449",
"1742",
"7269",
"32433",
"153850",
"772397",
"4088773",
"22746858",
"132601933",
"807880821",
"5132235182",
"33925263901",
"232905588441",
"1657807491222",
"12215424018837",
"93042845392105",
"731622663432978",
"5931915237693517",
"49535826242154973"
] | Sum of upward diagonals of Eulerian triangle. |
Subsets and Splits