A291698 -id:A291698 - cp's OEIS frontend

A292419 a(n) = [x^n] Product_{k>=1} (1 + nx^k) / (1 - nx^k).

1, 2, 12, 96, 872, 9960, 138180, 2298016, 44686224, 995739498, 24993249820, 697309946784, 21396151468536, 715827315312200, 25926440773118340, 1010478298772398080, 42162515927954808352, 1875027040759682964144, 88527520717734462201756, 4422273966757678408594560

Offset: 0

Vaclav Kotesovec, Sep 16 2017

nonn

Convolution of A291698 and A124577.

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A124577, A291698, A292418.

nmax = 25; Table[SeriesCoefficient[Product[(1+n*x^k)/(1-n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

{a(n)= polcoef(prod(k=1, n, ((1+n*x^k)/(1-n*x^k) +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

a(n) ~ 2 * n^n * (1 + 2/n + 4/n^2 + 8/n^3 + 14/n^4 + 24/n^5 + 40/n^6 + 64/n^7 + 100/n^8 + 154/n^9 + 232/n^10), for coefficients see A015128.

A300411 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - a(k)*x^a(k)).

Original entry on oeis.org

1, 1, 1, 4, 9, 14, 19, 24, 45, 75, 105, 135, 229, 359, 503, 647, 1047, 1591, 2272, 2972, 4696, 6996, 9844, 12894, 20064, 29538, 41204, 54407, 84457, 123723, 171757, 225939, 348643, 508693, 703815, 923529, 1423892, 2076942, 2870977, 3763380, 5778379, 8414332, 11621307

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Cf. A006906, A093637, A151945, A196545, A291698, A293806, A293807, A296387.

a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - a[k] x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 42}]

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = -x - 2*x^2 + Product_{n>=1} 1/(1 - a(n)*x^a(n)).

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).

Original entry on oeis.org

1, 2, 5, 19, 49, 126, 469, 1177, 2881, 6481, 23101, 53725, 127153, 274288, 581925, 1860751, 4155649, 9279791, 19409221, 39839239, 77052401, 229393207, 481747949, 1035561408, 2082441025, 4153434376, 7822058869, 14686515649, 39394280689, 79657493191, 163600884901

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

Index entries for sequences related to partitions

Cf. A286957, A291698, A303070, A303071, A303914.

Table[SeriesCoefficient[1/(1 - x) Product[(1 + n x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/(1 - x) Exp[Sum[(-1)^(k + 1) n^k x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[QPochhammer[-n, x]/((1 + n) (1 - x)), {x, 0, n}], {n, 0, 30}]

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

a(n) = Sum_{j=0..n} A286957(j,n).

cp's OEIS Frontend

A292419 a(n) = [x^n] Product_{k>=1} (1 + nx^k) / (1 - nx^k).

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A300411 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - a(k)*x^a(k)).

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Author

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Crossrefs

Programs

Mathematica

Formula

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A292419 a(n) = [x^n] Product_{k>=1} (1 + n*x^k) / (1 - n*x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A300411 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - a(k)*x^a(k)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A304782 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + n*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A292419 a(n) = [x^n] Product_{k>=1} (1 + nx^k) / (1 - nx^k).

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).