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 1-3 of 3 results.

A356572 Expansion of e.g.f. sinh( (exp(3*x) - 1)/3 ).

Original entry on oeis.org

0, 1, 3, 10, 45, 307, 2718, 26371, 265359, 2778976, 30916863, 372113623, 4873075056, 68908186765, 1037694932823, 16438615126282, 271972422548361, 4687666317874495, 84181305836224422, 1576083180118379527, 30757003280682603699, 624671260245315540568
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Cf. A009599, A024429, A357617.
Cf. A357649.

Programs

  • Mathematica
    With[{m = 20}, Range[0, m]! * CoefficientList[Series[Sinh[(Exp[3*x] - 1)/3], {x, 0, m}], x]] (* Amiram Eldar, Oct 07 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh((exp(3*x)-1)/3))))
  • PARI
    a(n) = sum(k=0, (n-1)\2, 3^(n-1-2*k)*stirling(n, 2*k+1, 2));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 3^(n-1-2*k) * Stirling2(n,2*k+1).
a(n) ~ 3^n * exp(n/LambertW(3*n) - n - 1/3) * n^n / (LambertW(3*n)^n * 2*sqrt(1 + LambertW(3*n))). - Vaclav Kotesovec, Oct 07 2022

A357650 Expansion of e.g.f. cosh( (exp(4*x) - 1)/4 ).

Original entry on oeis.org

1, 0, 1, 12, 113, 1000, 8977, 86996, 959905, 12303888, 179038689, 2840696540, 47684181393, 835731314808, 15277172343409, 292597596283684, 5900038421042753, 125488177929542944, 2809541905807203009, 65903118624174027436, 1610968753088423886257
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Cf. A009153, A024430, A357649.
Cf. A357617.

Programs

  • Mathematica
    With[{m = 20}, Range[0, m]! * CoefficientList[Series[Cosh[(Exp[4*x] - 1)/4], {x, 0, m}], x]] (* Amiram Eldar, Oct 07 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(cosh((exp(4*x)-1)/4)))
  • PARI
    a(n) = sum(k=0, n\2, 4^(n-2*k)*stirling(n, 2*k, 2));

Formula

a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k) * Stirling2(n,2*k).
a(n) ~ 2^(2*n-1) * exp(n/LambertW(4*n) - n - 1/4) * n^n / (LambertW(4*n)^n * sqrt(1 + LambertW(4*n))). - Vaclav Kotesovec, Oct 07 2022

A357666 Expansion of e.g.f. sinh( (exp(4*x) - 1)/2 )/2.

Original entry on oeis.org

0, 1, 4, 20, 160, 1872, 25024, 348224, 5055488, 78571776, 1332573184, 24695206912, 493816963072, 10492449771520, 234399640633344, 5480635606908928, 134015043318054912, 3427700843478056960, 91642829715498336256, 2556218693498006929408
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Cf. A024429, A357664, A357665.
Cf. A357598, A357617, A357663.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh((exp(4*x)-1)/2)/2)))
  • PARI
    a(n) = sum(k=0, (n-1)\2, 4^(n-1-k)*stirling(n, 2*k+1, 2));

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} 4^(n-1-k) * Stirling2(n,2*k+1).
Showing 1-3 of 3 results.

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

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