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 1-4 of 4 results.

A380228 Expansion of e.g.f. exp( exp( (exp(2*x)-1)/2 ) - 1 ).

Original entry on oeis.org

1, 1, 4, 21, 139, 1108, 10287, 108699, 1285228, 16783395, 239571125, 3706900992, 61746357449, 1100827515921, 20902202270580, 420903243601485, 8955301860908499, 200664408693149164, 4721693823656357303, 116370390417335016731, 2997078741899026174972, 80492590654279893652283
Offset: 0

Views

Author

Seiichi Manyama, Jan 17 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(exp((exp(2*x)-1)/2)-1)))

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * Stirling2(n,k) * Bell(k).

A346433 E.g.f.: 1 / (2 - exp(2*(exp(x) - 1))).

Original entry on oeis.org

1, 2, 14, 142, 1910, 32094, 647126, 15223198, 409276054, 12378827166, 416006542550, 15378483225758, 620176642174742, 27094392220198814, 1274759052849262422, 64259896197635471006, 3455259407744574799254, 197401403111903906001310, 11941074177046918285056470
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 17 2021

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(2 - Exp[2 (Exp[x]- 1)]), {x, 0, nmax}], x] Range[0, nmax]!
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
    Table[Sum[StirlingS2[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - exp(2*(exp(x) - 1))))) \\ Michel Marcus, Jul 18 2021

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A001861(k) * a(n-k).
a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^k * A000670(k).
a(n) ~ n! / (2*(2+log(2)) * (log(1+log(2)/2))^(n+1)). - Vaclav Kotesovec, Jul 27 2021

A369783 Expansion of e.g.f. exp( exp(3*(exp(x)-1))-1 ).

Original entry on oeis.org

1, 3, 21, 192, 2154, 28434, 429213, 7261788, 135698268, 2769463335, 61186736415, 1452889463034, 36857766745749, 993941679586098, 28370018078000985, 853903169641805925, 27014392815958815969, 895723118730738795837, 31048284069527339602902
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(exp(3*(exp(x)-1))-1)))

Formula

a(n) = Sum_{k=0..n} 3^k * Stirling2(n,k) * Bell(k) = Sum_{k=0..n} Stirling2(n,k) * A247452(k).

A369784 Expansion of e.g.f. exp( (exp(2*(exp(x)-1))-1)/2 ).

Original entry on oeis.org

1, 1, 4, 21, 137, 1068, 9663, 99249, 1137858, 14373531, 198031153, 2951536030, 47270242621, 808917666365, 14720125466652, 283667520561633, 5768057979319853, 123364873473674732, 2767400573883314755, 64950007415991458989, 1591227433994704322322
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(2*(exp(x)-1))-1)/2)))

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A004211(k).
Showing 1-4 of 4 results.