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.

A354241 Expansion of e.g.f. 1/sqrt(1 + 4 * log(1-x)).

Original entry on oeis.org

1, 2, 14, 160, 2544, 51888, 1292208, 38012448, 1289847456, 49593778368, 2130914229312, 101188640375040, 5262325852773120, 297450338175682560, 18157597034693207040, 1190483599149657584640, 83433723762978141189120, 6224485980052510972692480
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A354240, A354242, A354244.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+4*log(1-x))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(-log(1-x))^k)))
  • PARI
    a(n) = sum(k=0, n, (2*k)!*abs(stirling(n, k, 1))/k!);

Formula

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (-log(1-x))^k.
a(n) = Sum_{k=0..n} (2*k)! * |Stirling1(n,k)|/k!.
a(n) ~ n^n / (sqrt(2) * (exp(1/4)-1)^(n + 1/2) * exp(3*n/4)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

A354243 Expansion of e.g.f. Sum_{k>=0} (2*k)! * log(1+x)^k / k!.

Original entry on oeis.org

1, 2, 22, 652, 36252, 3249648, 427841136, 77725790784, 18629187576192, 5694658698037824, 2162203542669622464, 998275836346954738560, 550745779092109449586560, 357819370067278253918223360, 270404811566689476740771496960
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A006252, A354250.
Cf. A316747, A354240, A354244.

Programs

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

Formula

a(n) = Sum_{k=0..n} (2*k)! * Stirling1(n,k).

A354251 Expansion of e.g.f. Sum_{k>=0} (3*k)! * (-log(1-x))^k / k!.

Original entry on oeis.org

1, 6, 726, 365052, 481186836, 1312477120944, 6422029618230000, 51225621215200895520, 621881012244669445985760, 10911233517605729917096273920, 265743399210784245852461349120000, 8697920910678436598411074217669652480
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A007840, A354244.
Cf. A316748, A353118, A354250.

Programs

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

Formula

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

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

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