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 1-7 of 7 results.

A379943 Expansion of e.g.f. 1/( exp(-x) - x )^4.

Original entry on oeis.org

1, 8, 76, 844, 10776, 155844, 2520856, 45125924, 886037216, 18938440324, 437820992136, 10886467502244, 289738784758096, 8218731027307844, 247539834718198136, 7889896358130120484, 265325716114102815936, 9388476560982511842564, 348703400008471862936296
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A072597, A379933, A379942.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^4))
  • PARI
    a(n) = n!*sum(k=0, n, (k+4)^(n-k)*binomial(k+3, 3)/(n-k)!);

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A072597.
a(n) = n! * Sum_{k=0..n} (k+4)^(n-k) * binomial(k+3,3)/(n-k)!.

A379942 Expansion of e.g.f. 1/( exp(-x) - x )^3.

Original entry on oeis.org

1, 6, 45, 411, 4449, 55803, 796581, 12757503, 226588257, 4420898595, 94001021589, 2163619250895, 53598352999905, 1421924243354787, 40221778417553637, 1208471542554184767, 38434396264371831873, 1289995362325669726659, 45567027291743788320405
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A072597, A379933, A379943.
Cf. A377530.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^3))
  • PARI
    a(n) = n!*sum(k=0, n, (k+3)^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A072597.
a(n) = n! * Sum_{k=0..n} (k+3)^(n-k) * binomial(k+2,2)/(n-k)!.

A379936 E.g.f. A(x) satisfies A(x) = 1/( exp(-x*A(x)^(1/2)) - x )^2.

Original entry on oeis.org

1, 4, 30, 344, 5400, 108492, 2667952, 77811120, 2629399680, 101122817300, 4363964377344, 208925612290056, 10992411683169280, 630611992509716700, 39182624685283891200, 2621745777377998537568, 187969244952968687812608, 14377545994804829244970020
Offset: 0

Views

Author

Seiichi Manyama, Jan 06 2025

Keywords

Crossrefs

Cf. A379933, A379934.
Cf. A088690.

Programs

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

Formula

E.g.f.: ( (1/x) * Series_Reversion( x*exp(-x)/(1+x) ) )^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A088690.
a(n) = 2 * n! * Sum_{k=0..n} (n+2)^(k-1) * binomial(n+2,n-k)/k!.

A379934 E.g.f. A(x) satisfies A(x) = 1/( exp(-x*A(x)) - x )^2.

Original entry on oeis.org

1, 4, 38, 626, 15008, 476122, 18864124, 898099526, 49988162672, 3187006372466, 229091274174404, 18335328399262030, 1617287276785929928, 155893591123924724618, 16304903025947743812476, 1839154613521698544945238, 222562344165125395485931232, 28763041177430039602579211746
Offset: 0

Views

Author

Seiichi Manyama, Jan 06 2025

Keywords

Crossrefs

Cf. A379933, A379936.
Cf. A379864, A379884.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A379884.
a(n) = 2 * n! * Sum_{k=0..n} (n+k+2)^(k-1) * binomial(n+k+2,n-k)/k!.

A379992 Expansion of e.g.f. exp(-3*x)/(exp(-x) - x)^2.

Original entry on oeis.org

1, 1, 7, 41, 349, 3539, 42451, 585605, 9130297, 158692679, 3041499871, 63712004729, 1447946191957, 35479218963083, 932326476195115, 26153289728300909, 779995883104560241, 24644267406802467215, 822278654588440803511, 28891372907012629446881
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A358738, A377529, A379933, A379997.
Cf. A368266, A377530, A379994.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[-3x]/(Exp[-x]-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 14 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)*(k-1)^(n-k)/(n-k)!);

Formula

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

A379997 Expansion of e.g.f. 1/(exp(x) - x*exp(2*x))^2.

Original entry on oeis.org

1, 0, 6, 22, 224, 2138, 25732, 351846, 5458224, 94441042, 1803255404, 37652268014, 853321021192, 20858236815258, 546941712302052, 15313467390967222, 455933682027961184, 14383416438784605602, 479254037890010238172, 16817855455956128823486, 619953003446894086537656
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A358738, A377529, A379933, A379992.
Cf. A092148.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A092148.
a(n) = n! * Sum_{k=0..n} (k+1) * (k-2)^(n-k)/(n-k)!.

A379935 E.g.f. A(x) satisfies A(x) = 1/( exp(-x) - x*A(x) )^2.

Original entry on oeis.org

1, 4, 38, 674, 17744, 623362, 27480844, 1460031610, 90862627184, 6485745312098, 522469881832964, 46895105170999978, 4641403797239576392, 502226056825606487506, 58985555898802967473820, 7473459685930447455067418, 1016083115772085962460442336, 147559760656716707828287356610
Offset: 0

Views

Author

Seiichi Manyama, Jan 06 2025

Keywords

Crossrefs

Cf. A379864, A379886, A379933.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A379886.
a(n) = 2 * n! * Sum_{k=0..n} (3*n-3*k+2)^(k-1) * binomial(3*n-3*k+2,n-k)/k!.
Showing 1-7 of 7 results.

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