A316087 -id:A316087 - cp's OEIS frontend

A316088 Expansion of 1/(1 + Sum_{k>=1} k^3 * x^k).

1, -1, -7, -12, 31, 193, 240, -1105, -5167, -3924, 36343, 133873, 31584, -1131025, -3343639, 1240212, 33732367, 79895089, -90574128, -970716385, -1800454975, 3954181452, 27045519079, 37164094177, -145299908928, -730358292769, -653629025575, 4869632030004

Offset: 0

Seiichi Manyama, Jun 24 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..2000

1/(1+ Sum_{k>=1} k^m * x^k): A163810 (m=1), A316087 (m=2), this sequence (m=3).

N=99; x='x+O('x^N); Vec((x-1)^4/(x^4-3*x^3+10*x^2-3*x+1))

G.f.: (x-1)^4/(x^4-3*x^3+10*x^2-3*x+1).

a(0) = 1; a(n) = -Sum_{k=1..n} k^3 * a(n-k). - Ilya Gutkovskiy, Feb 02 2021

A316086 Product_{k>=1} 1/(1 - a(k)x^k) = 1 + Sum_{k>=1} k^2x^k.

Original entry on oeis.org

1, 3, 5, -2, -6, -35, -4, 8, 16, -62, -158, -1149, 56, 556, 680, -1566, -4966, 3544, 6132, 20268, 15792, -75194, -162814, -1246153, 333904, 805112, 235968, -2775034, -5259270, 7160120, 15199212, 25457976, -4403208, -119142438, -161039518, 211525304, 635914664

Offset: 1

Seiichi Manyama, Jun 24 2018

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			1/((1-x)*(1-3*x^2)*(1-5*x^3)*(1+2*x^4)* ... ) = 1 + x + 4*x^2 + 9*x^3 + 16*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A147559, A253909, A316087.

A335577 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).

Original entry on oeis.org

1, -1, -2, 9, 32, -285, -1236, 18725, 86176, -2087001, -9204580, 351964569, 1336442304, -83422970917, -231889447076, 26389118293005, 35917342192064, -10722110983670193, 5028963509133756, 5432569724760331841, -14852185163192897120, -3352369390318855889661

Offset: 0

Ilya Gutkovskiy, Jan 26 2021

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Cf. A302189, A302397, A308861, A316087, A335578.

a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] k^2 a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]
nmax = 21; CoefficientList[Series[1/(1 + Exp[x] x (1 + x)), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (1 + exp(x) * x * (1 + x)).

E.g.f.: 1 / (1 + Sum_{k>=1} k^2 * 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

Ilya Gutkovskiy, Oct 08 2018

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Cf. A301655, A316087, A316088.

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}]

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

cp's OEIS Frontend

A316088 Expansion of 1/(1 + Sum_{k>=1} k^3 * x^k).

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A316086 Product_{k>=1} 1/(1 - a(k)x^k) = 1 + Sum_{k>=1} k^2x^k.

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A335577 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).

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Author

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Programs

Mathematica

Formula

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

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Author

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Mathematica

Formula

cp's OEIS Frontend

A316088 Expansion of 1/(1 + Sum_{k>=1} k^3 * x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A316086 Product_{k>=1} 1/(1 - a(k)*x^k) = 1 + Sum_{k>=1} k^2*x^k.

Views

Author

Keywords

Examples

Links

Crossrefs

A335577 a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * k^2 * a(n-k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A316086 Product_{k>=1} 1/(1 - a(k)x^k) = 1 + Sum_{k>=1} k^2x^k.