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.

A377546 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^2 ).

Original entry on oeis.org

1, 2, 18, 294, 7136, 231410, 9421932, 462459242, 26593896912, 1754278123266, 130611457831700, 10835721949072922, 991315043401627320, 99154012317212577218, 10765112531819005907484, 1260860266373297376720810, 158473050112495481401395872, 21275613503385328981848681986
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Links

Crossrefs

Cf. A213644, A377548.
Cf. A364985.

Programs

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

Formula

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364985.
a(n) = 2 * n! * Sum_{k=0..n} k^(n-k) * binomial(2*n+k+2,k)/( (2*n+k+2)*(n-k)! ).

A377629 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^4 ).

Original entry on oeis.org

1, 4, 60, 1644, 66712, 3611620, 245284344, 20071928212, 1923688610400, 211438912978692, 26225665058289640, 3624147718351890004, 552229557439437084816, 91990834731657653530180, 16632301623786709606057368, 3243982650658692575922907860, 678932992008068232965498759104
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Links

Crossrefs

Cf. A213644, A377546, A377548.
Cf. A377631, A377632.

Programs

  • PARI
    a(n) = 4*n!*sum(k=0, n, k^(n-k)*binomial(4*n+k+4, k)/((4*n+k+4)*(n-k)!));

Formula

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377631.
a(n) = 4 * n! * Sum_{k=0..n} k^(n-k) * binomial(4*n+k+4,k)/( (4*n+k+4)*(n-k)! ).

A382059 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 127, 3733, 152161, 7939261, 505087843, 37920697753, 3281899787137, 321700411900441, 35227497466867531, 4262151791317099285, 564639582580738851265, 81290104199287214904037, 12637400195063381931755731, 2109868901338065949399370161, 376504852688521502050554789889
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A161629, A382058.
Cf. A364938, A382033.
Cf. A002293, A377548, A382034.

Programs

  • PARI
    a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, (k+1)^(n-k-1)*binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));

Formula

Let F(x) be the e.g.f. of A377548. F(x) = log(A(x))/x = B(x*A(x))^3.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^3 ) ).
a(n) = 3 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
Showing 1-3 of 3 results.

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

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