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.

A356672 a(n) = n! * Sum_{k=0..n} k^(2*(n-k))/k!.

Original entry on oeis.org

1, 1, 3, 19, 253, 5661, 188191, 8983423, 594848409, 52174034713, 5852229698971, 822684190381131, 142739480367287893, 30074750245383836149, 7575373641076070706423, 2252600759590927171373431, 783103569459739402827046321, 315587346190678252431713684913
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2022

Keywords

Crossrefs

Programs

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

Formula

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

A356688 a(n) = n! * Sum_{k=0..n} k^(3*n)/k!.

Original entry on oeis.org

1, 1, 66, 21225, 18952156, 36175231585, 126556309395486, 733064060959310689, 6540867625730306094360, 85180334386943946887707617, 1552697061493449955344530003290, 38315904135534199560725372265381721, 1245605749857294018587318829355458646068
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := n! * Sum[k^(3*n)/k!, {k, 0, n}]; a[0] = 1; Array[a, 13, 0] (* Amiram Eldar, Aug 23 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(3*n)/k!);
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^3*x)^k/(k!*(1-k^3*x)))))

Formula

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

A358687 a(n) = n! * Sum_{k=0..n} k^(3 * (n-k)) / (n-k)!.

Original entry on oeis.org

1, 1, 4, 57, 1444, 61785, 4050126, 373648513, 47101090744, 7764843893265, 1630744323319450, 426925697290933401, 136591846585403311620, 52602030074554601172649, 24058544668572618782040022, 12916480280574798627072144465
Offset: 0

Views

Author

Seiichi Manyama, Nov 26 2022

Keywords

Crossrefs

Programs

  • Mathematica
    Join[{1}, Table[n! * Sum[k^(3*(n-k))/(n-k)!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 27 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(3*(n-k))/(n-k)!);
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k*exp(x)^k^3)))
    
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k/(1-k^3*x)^(k+1)))

Formula

E.g.f.: Sum_{k>=0} x^k * exp(k^3 * x).
G.f.: Sum_{k>=0} k! * x^k / (1 - k^3 * x)^(k+1).
log(a(n)) ~ (6*n*(log(n) - 1) + 3*log(1 + LambertW(n^(2/3))) + 2*n*LambertW(n^(2/3)) * (7*log(n) - 6*log(1 + LambertW(n^(2/3))) + 3*LambertW(n^(2/3)))) / (6*(1 + LambertW(n^(2/3)))). - Vaclav Kotesovec, Nov 27 2022
Showing 1-4 of 4 results.