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.

A353226 Expansion of e.g.f. (1 - x^2)^(-x).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 360, 1680, 20160, 151200, 1663200, 17962560, 219542400, 2854051200, 40441040640, 606356150400, 9793028044800, 166481476761600, 3017626733721600, 57359043873331200, 1153275200453376000, 24233844054131712000, 535361100608439705600
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Crossrefs

Cf. A066166, A121452, A351155, A353227.

Programs

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

Formula

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

A375167 Expansion of e.g.f. 1 / (1 + x * log(1 - x^2/2)).

Original entry on oeis.org

1, 0, 0, 3, 0, 15, 180, 210, 5040, 51030, 207900, 3991680, 42411600, 356756400, 6485398920, 80635054500, 1040690851200, 19440077857200, 291313362740400, 4914773560897200, 98182334033784000, 1763213788027692000, 35636304386103220800, 778379605589616030000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052830, A375556, A375558.
Cf. A351155.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x*log(1-x^2/2))))
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k)!*abs(stirling(k, n-2*k, 1))/(2^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)! * |Stirling1(k,n-2*k)|/(2^k*k!).

A351156 Expansion of e.g.f. (1 - x^3/6)^(-x).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 70, 560, 0, 5600, 92400, 369600, 1201200, 30830800, 252252000, 1210809600, 19059040000, 240143904000, 1738184448000, 22451549120000, 342205063200000, 3417705170880000, 43866126368064000, 732641268463104000, 9234973972224000000
Offset: 0

Views

Author

Seiichi Manyama, May 02 2022

Keywords

Crossrefs

Cf. A351155, A353227.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3/6)^(-x)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^3/6))))
  • 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)*6^(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))/(6^k*k!));

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+2)/3)} (3*k-2)/((k-1) * 6^(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)|/(6^k*k!).
a(n) ~ sqrt(2*Pi) * n^(n - 1/2 + 6^(1/3)) / (Gamma(6^(1/3)) * 3^(6^(1/3)) * exp(n) * 6^(n/3)). - Vaclav Kotesovec, May 04 2022
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.