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.

A316087 Expansion of 1/(1 + Sum_{k>=1} k^2 * x^k).

Original entry on oeis.org

1, -1, -3, -2, 7, 19, 8, -53, -119, -18, 387, 727, -112, -2745, -4315, 2238, 18991, 24715, -24296, -128461, -135023, 219502, 850635, 688239, -1806560, -5515441, -3116403, 14022398, 34994967, 10783939, -104389592, -216919973, -5497639, 752295022, 1309660627
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2018

Keywords

Links

Crossrefs

1/(1+ Sum_{k>=1} k^m * x^k): A163810 (m=1), this sequence (m=2), A316088 (m=3).
Cf. A253909, A316086.

Programs

  • PARI
    N=99; x='x+O('x^N); Vec(1/(1+sum(k=1, sqrtint(N), k^2*x^k)))

Formula

Convolution inverse of A253909.
G.f.: (x-1)^3/(x^3-4*x^2+2*x-1).
a(0) = 1; a(n) = -Sum_{k=1..n} k^2 * a(n-k). - Ilya Gutkovskiy, Feb 02 2021

A335578 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^3 * a(n-k).

Original entry on oeis.org

1, -1, -6, 15, 272, -745, -29976, 61271, 6065856, -2723697, -1941455080, -3989345041, 897021218400, 4964061925511, -562221881675832, -5689641396555705, 456732442022509184, 7321841133968133023, -464200472167634521800, -10961686347887871324289, 573373115861405030522400
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 26 2021

Keywords

Crossrefs

Cf. A302397, A302870, A308862, A316088, A335577.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^3 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[1/(1 + Exp[x] x (1 + 3 x + x^2)), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (1 + exp(x) * x * (1 + 3*x + x^2)).
E.g.f.: 1 / (1 + Sum_{k>=1} k^3 * x^k / k!).

A320257 a(n) = [x^n] 1/(1 + Sum_{k>=1} k^n*x^k).

Original entry on oeis.org

1, -1, -3, -12, 115, 10801, 513128, -31622305, -29139002795, -10674262825020, 8501486817673047, 49739414341982655313, 119603027548636893292304, -1457935338988787721752835601, -53309412371631686621966575526695, -722987647385617983636099803915844108
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 08 2018

Keywords

Crossrefs

Cf. A301655, A316087, A316088.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 + Sum[k^n x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 + PolyLog[-n, x]), {x, 0, n}], {n, 0, 15}]

Formula

a(n) = [x^n] 1/(1 + PolyLog(-n,x)), where PolyLog() is the polylogarithm function.
Showing 1-3 of 3 results.

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

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