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.

A380646 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x)/(1 + x)^2 ).

Original entry on oeis.org

1, 4, 46, 932, 27568, 1080432, 52916176, 3115326496, 214470890496, 16914853191680, 1504252282653184, 148956086481767424, 16256865070022066176, 1938988214539948730368, 250943399365390735104000, 35026523834624205803491328, 5245178283068781060488298496, 838841884254236846183525646336
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2025

Keywords

Links

Crossrefs

Cf. A088690, A380647, A380648.
Cf. A065866, A377829.
Cf. A097629, A380828.
Cf. A377892.

Programs

  • Mathematica
    nmax=18; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-2*x]/(1 + x)^2 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)
  • PARI
    a(n) = 2*n!*sum(k=0, n, (2*n+2)^(k-1)*binomial(2*n+2, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * exp(2 * x * A(x)).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377892.
a(n) = 2 * n! * Sum_{k=0..n} (2*n+2)^(k-1) * binomial(2*n+2,n-k)/k!.

A380830 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x) ).

Original entry on oeis.org

1, 4, 47, 978, 29769, 1201728, 60656679, 3681441648, 261337079601, 21256149703680, 1949700750690879, 199146039242552064, 22420399033075845177, 2758645779752490872832, 368321963942753147683575, 53038788218443786432223232, 8194316429830951008255159009, 1352065789150879084276947222528
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2025

Keywords

Links

Crossrefs

Cf. A361182, A380826, A380829.
Cf. A088690, A380828.
Cf. A376094.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(3*x*A(x)) / ( 1 - x*exp(3*x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (3*n+3)^k * binomial(n,k)/(k+1)!.
a(n) = A376094(n+1)/(n+1).

A380808 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x) / (1 + x*exp(-x)) ).

Original entry on oeis.org

1, 3, 24, 335, 6812, 183397, 6168406, 249350285, 11785793352, 638146503593, 38960123581154, 2648475653518081, 198429466488527164, 16246940820392924189, 1443430758561178861758, 138305198841617791230533, 14217431594874334746229520, 1560842183273111251153540945
Offset: 0

Views

Author

Seiichi Manyama, Feb 04 2025

Keywords

Links

Crossrefs

Cf. A088690, A161633, A380826.
Cf. A352448, A380828.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+k+2)^k*binomial(n, k)/(k+1)!);

Formula

E.g.f. A(x) satisfies A(x) = exp(2*x*A(x)) / ( 1 - x*exp(x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (n+k+2)^k * binomial(n,k)/(k+1)!.
Showing 1-3 of 3 results.

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

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