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.

A356673 a(n) = n! * Sum_{k=0..n} k^(3*(n-k))/k!.

Original entry on oeis.org

1, 1, 3, 31, 901, 45741, 3960871, 584698843, 130554106761, 40790044059481, 17681098707667531, 10491554658622447191, 8198225417359164798733, 8172446419302496167191941, 10264848632098736708582150511
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2022

Keywords

Crossrefs

Programs

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

Formula

E.g.f: Sum_{k>=0} x^k / (k! * (1 - k^3 * x)).

A356674 a(n) = n! * Sum_{k=0..n} k^(k*(n-k))/k!.

Original entry on oeis.org

1, 2, 5, 25, 349, 19941, 4440391, 4382699203, 17687865017481, 356274213630958297, 33338407933090938442411, 16214021627369697901867402911, 43817834057167927861655409052462093, 595284492835035398061242850538179192931525
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2022

Keywords

Crossrefs

Programs

  • Mathematica
    Table[n!*(1 + Sum[k^(k*(n-k))/k!, {k, 1, n}]), {n, 0, 12}] (* Vaclav Kotesovec, Nov 27 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(k*(n-k))/k!);
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k^k*x)))))

Formula

E.g.f: Sum_{k>=0} x^k / (k! * (1 - k^k * x)).
log(a(n)) ~ n^2*log(n)/4 * (1 - log(2)/log(n) + 1/(4*log(n)^2)). - Vaclav Kotesovec, Nov 27 2022

A356687 a(n) = n! * Sum_{k=0..n} k^(2*n)/k!.

Original entry on oeis.org

1, 1, 18, 927, 94876, 16251045, 4210190766, 1543550310211, 764096247603480, 493254380867214249, 404269328278061434810, 411862088865696890314311, 512690851568229926690616948, 768775988931240685277619894157
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := n! * Sum[k^(2*n)/k!, {k, 0, n}]; a[0] = 1; Array[a, 14, 0] (* Amiram Eldar, Aug 23 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(2*n)/k!);
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*x)^k/(k!*(1-k^2*x)))))

Formula

E.g.f.: Sum_{k>=0} (k^2 * x)^k / (k! * (1 - k^2 * x)).
Showing 1-3 of 3 results.