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.

A353227 Expansion of e.g.f. (1 - x^3)^(-x).

Original entry on oeis.org

1, 0, 0, 0, 24, 0, 0, 2520, 20160, 0, 1209600, 19958400, 79833600, 1556755200, 39956716800, 326918592000, 5056340889600, 148203095040000, 1867358997504000, 30411275102208000, 946128558735360000, 15965919428659200000, 293266062902292480000
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A066166, A351156, A353223, A353226.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)^(-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^3))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, (i+2)\3, (3*j-2)/(j-1)*v[i-3*j+3]/(i-3*j+2)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, abs(stirling(k, n-3*k, 1))/k!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+2)/3)} (3*k-2)/(k-1) * a(n-3*k+2)/(n-3*k+2)!.
a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(k,n-3*k)|/k!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3*exp(n)). - Vaclav Kotesovec, May 04 2022

A351155 Expansion of e.g.f. (1 - x^2/2)^(-x).

Original entry on oeis.org

1, 0, 0, 3, 0, 15, 90, 210, 2520, 13230, 103950, 873180, 7484400, 72972900, 745404660, 8185126950, 95805309600, 1184852869200, 15538995271800, 214159261516200, 3109622647131000, 47252530639314000, 752635500963746400, 12499951421009052000, 216709136059079664000
Offset: 0

Views

Author

Seiichi Manyama, May 02 2022

Keywords

Crossrefs

Cf. A351156, A353226.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^2/2)^(-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^2/2))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, (i+1)\2, (2*j-1)/((j-1)*2^(j-1))*v[i-2*j+2]/(i-2*j+1)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, abs(stirling(k, n-2*k, 1))/(2^k*k!));

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+1)/2)} (2*k-1)/((k-1) * 2^(k-1)) * a(n-2*k+1)/(n-2*k+1)!.
a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(k,n-2*k)|/(2^k*k!).
a(n) ~ sqrt(Pi) * n^(n - 1/2 + sqrt(2)) / (Gamma(sqrt(2)) * exp(n) * 2^(n/2 + sqrt(2) - 1/2)). - Vaclav Kotesovec, May 04 2022

A375556 Expansion of e.g.f. 1 / (1 + x * log(1 - x^3/6)).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 70, 1120, 0, 5600, 184800, 2217600, 1201200, 61661600, 1513512000, 16682265600, 38118080000, 1440863424000, 31721866176000, 352561745536000, 2053230379200000, 68832104140800000, 1449890913639168000, 17583390443114496000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052830, A375167, A375558.
Cf. A351156.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x*log(1-x^3/6))))
  • PARI
    a(n) = n!*sum(k=0, n\3, (n-3*k)!*abs(stirling(k, n-3*k, 1))/(6^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)! * |Stirling1(k,n-3*k)|/(6^k*k!).
Showing 1-3 of 3 results.

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