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.

A357649 Expansion of e.g.f. cosh( (exp(3*x) - 1)/3 ).

Original entry on oeis.org

1, 0, 1, 9, 64, 435, 3097, 24822, 232759, 2517345, 30070954, 382827225, 5110770205, 71421582024, 1049487311485, 16286699945853, 267145966335088, 4616924929100535, 83622792656855125, 1578916985654901366, 30957723637379211115, 628927539690331202661
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Programs

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

Formula

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

A357617 Expansion of e.g.f. sinh( (exp(4*x) - 1)/4 ).

Original entry on oeis.org

0, 1, 4, 17, 88, 657, 6844, 83393, 1072880, 14242785, 197046964, 2895895345, 45930435016, 789930042865, 14628150636012, 287915593953889, 5950831121362656, 128180962018224833, 2868724306984850020, 66704877850797014353, 1613138176448134032440
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Programs

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

Formula

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

A357663 Expansion of e.g.f. cosh( (exp(4*x) - 1)/2 ).

Original entry on oeis.org

1, 0, 4, 48, 464, 4480, 48448, 621824, 9320704, 154890240, 2746131456, 51237908480, 1007228375040, 20965557829632, 463091379159040, 10826828061147136, 266438312153120768, 6861616219559034880, 184128217520198123520, 5135753969867535941632
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2022

Keywords

Crossrefs

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Cosh[(Exp[4x]-1)/2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 13 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(cosh((exp(4*x)-1)/2)))
    
  • PARI
    a(n) = sum(k=0, n\2, 4^(n-k)*stirling(n, 2*k, 2));

Formula

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