cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A367135 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^3)).

Original entry on oeis.org

1, 1, 9, 166, 4719, 182326, 8927301, 529922002, 36988772211, 2969132797966, 269488306924833, 27291375956851546, 3050923148547318039, 373187615576953777510, 49580088565083198922845, 7109665420655116517351458, 1094492388113416460752513851
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A000670, A052894, A367134.
Cf. A367137, A367139.

Programs

  • Mathematica
    Table[1/(3*n+1)! * Sum[(3*n+k)! * StirlingS2[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (3*n+k)!*stirling(n, k, 2))/(3*n+1)!;

Formula

a(n) = (1/(3*n+1)!) * Sum_{k=0..n} (3*n+k)! * Stirling2(n,k).
a(n) ~ 3^(4*n) * LambertW(2*exp(1/3)/3)^(3*n + 1) * n^(n-1) / (sqrt(1 + LambertW(2*exp(1/3)/3)) * 2^(3*n + 1) * exp(n) * (3*LambertW(2*exp(1/3)/3) - 1)^(4*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367138 E.g.f. satisfies A(x) = 1/(1 + log(1 - x*A(x)^2)).

Original entry on oeis.org

1, 1, 7, 98, 2096, 60684, 2221766, 98488592, 5129567208, 307066395000, 20775900638472, 1567955813868960, 130596146677118448, 11899839375083061024, 1177540373453616858240, 125754589311488009416704, 14416305655742615673941760, 1765794816084642802179333120
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A007840, A052802, A367139.
Cf. A367134, A367136.

Programs

  • Mathematica
    Table[1/(2*n+1)! * Sum[(2*n+k)! * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (2*n+k)!*abs(stirling(n, k, 1)))/(2*n+1)!;

Formula

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} (2*n+k)! * |Stirling1(n,k)|.
a(n) ~ LambertW(2*exp(3))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(3)))) * exp(n) * (-2 + LambertW(2*exp(3)))^(3*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A367136 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x)^2)).

Original entry on oeis.org

1, 1, 5, 50, 764, 15804, 413426, 13094864, 487323000, 20844584760, 1007739144312, 54343954158240, 3234285062655984, 210581685526690464, 14889759832273000320, 1136236597054802033664, 93074880409847175490560, 8146156595011083708521472
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2023

Keywords

Crossrefs

Cf. A006252, A198860, A367137.
Cf. A367134, A367138.

Programs

  • Mathematica
    Table[1/(2*n+1)! * Sum[(2*n+k)! * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 07 2023 *)
  • PARI
    a(n) = sum(k=0, n, (2*n+k)!*stirling(n, k, 1))/(2*n+1)!;

Formula

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} (2*n+k)! * Stirling1(n,k).
a(n) ~ LambertW(2*exp(1))^n * n^(n-1) / (sqrt(2*(1 + LambertW(2*exp(1)))) * exp(n) * (2 - LambertW(2*exp(1)))^(3*n + 1)). - Vaclav Kotesovec, Nov 07 2023

A376389 Expansion of e.g.f. (1/x) * Series_Reversion( x*(2 - exp(x))^2 ).

Original entry on oeis.org

1, 2, 16, 236, 5172, 151452, 5568452, 246816236, 12817081828, 763506280700, 51333645252228, 3845783934171852, 317719919221661540, 28697779828343464412, 2813593953407672094724, 297587218343306095847084, 33775895041558685181041892, 4094844200848292606224524732
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Links

Crossrefs

Cf. A005649, A367134.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(2-exp(x))^2)/x))
  • PARI
    a(n) = 2*sum(k=0, n, (2*n+k+1)!*stirling(n, k, 2))/(2*n+2)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(2 - exp(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367134.
a(n) = (2/(2*n+2)!) * Sum_{k=0..n} (2*n+k+1)! * Stirling2(n,k).
a(n) ~ 2^n * LambertW(exp(1/2))^(2*n + 2)*n^(n-1) / (sqrt(1 + LambertW(exp(1/2))) * exp(n) * (2*LambertW(exp(1/2)) - 1)^(3*n + 2)). - Vaclav Kotesovec, Sep 22 2024

A377347 E.g.f. satisfies A(x) = 1 + (exp(x*A(x)^2) - 1)/A(x)^2.

Original entry on oeis.org

1, 1, 1, 7, 41, 391, 4509, 62847, 1038001, 19580071, 418681877, 9973993855, 262293996777, 7545559829991, 235715629493005, 7946944965054271, 287592204406672481, 11120005819664145895, 457514133092462477253, 19957535405566629526335, 920056233384401619083545
Offset: 0

Views

Author

Seiichi Manyama, Oct 26 2024

Keywords

Crossrefs

Cf. A052750, A367134, A367180, A377330.
Cf. A377326, A377348.

Programs

  • PARI
    a(n) = sum(k=0, (2*n+1)\3, (2*n-2*k)!/(2*n-3*k+1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..floor((2*n+1)/3)} (2*n-2*k)!/(2*n-3*k+1)! * Stirling2(n,k).

A377424 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^4)).

Original entry on oeis.org

1, 1, 11, 253, 9019, 438021, 26992707, 2018069341, 177498369419, 17959376607061, 2055112480694323, 262437681414074541, 36999068388057870651, 5708040382071000644581, 956533539112835413864739, 173022072326584494697760893, 33600521994423195247370822251, 6972639514725247888782370422261
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A000670, A052894, A367134, A367135.
Cf. A377425, A377428.

Programs

  • PARI
    a(n) = sum(k=0, n, (4*n+k)!*stirling(n, k, 2))/(4*n+1)!;

Formula

a(n) = (1/(4*n+1)!) * Sum_{k=0..n} (4*n+k)! * Stirling2(n,k).

A377425 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^2))^2.

Original entry on oeis.org

1, 2, 24, 572, 20788, 1021892, 63498116, 4776128772, 422019084132, 42854861672612, 4918270207805188, 629575456637707076, 88938171122678982692, 13744507646644260776292, 2306659049841490720035780, 417774877069420589127228164, 81222489094387608969950071780
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Cf. A367134, A376389.
Cf. A377424, A377428.

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (4*n+k+1)!*stirling(n, k, 2))/(4*n+2)!;

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377424.
a(n) = (2/(4*n+2)!) * Sum_{k=0..n} (4*n+k+1)! * Stirling2(n,k).

A371296 E.g.f. satisfies A(x) = 1/(3 - 2*exp(x*A(x)^2)).

Original entry on oeis.org

1, 2, 26, 674, 26682, 1429682, 96867178, 7946279490, 765861255002, 84837503946962, 10621798904563530, 1483378875680954210, 228626616449674796602, 38549099486166110798322, 7058696888173770772536362, 1394913467379909728350803074, 295904373562519633314958421274
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Crossrefs

Cf. A004123, A258922.
Cf. A367134.

Programs

  • PARI
    a(n) = sum(k=0, n, 2^k*(2*n+k)!*stirling(n, k, 2))/(2*n+1)!;

Formula

a(n) = (1/(2*n+1)!) * Sum_{k=0..n} 2^k * (2*n+k)! * Stirling2(n,k).

A377330 E.g.f. satisfies A(x) = 1 + A(x)^2 * (exp(x*A(x)^2) - 1).

Original entry on oeis.org

1, 1, 9, 163, 4537, 171451, 8206517, 476071275, 32469361617, 2546397256651, 225784275815485, 22336278201427675, 2439097416667718297, 291422424985108052091, 37817207428965579915333, 5296739332085114983427083, 796419825874139713780172449, 127955324543685857975407200235
Offset: 0

Views

Author

Seiichi Manyama, Oct 25 2024

Keywords

Crossrefs

Cf. A052750, A367134, A367180.

Programs

  • PARI
    a(n) = sum(k=0, n, (2*n+2*k)!/(2*n+k+1)!*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (2*n+2*k)!/(2*n+k+1)! * Stirling2(n,k).
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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