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A305841 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001970 (partitions of partitions). 2
1, 3, 3, 8, 7, 14, 15, 30, 30, 49, 56, 91, 101, 150, 176, 261, 297, 415, 490, 676, 792, 1058, 1255, 1666, 1958, 2537, 3010, 3868, 4565, 5780, 6842, 8610, 10143, 12607, 14883, 18392, 21637, 26505, 31185, 38014, 44583, 53966, 63261, 76233, 89134, 106813, 124754 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Inverse weigh transform of A001970.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..2000

N. J. A. Sloane, Transforms

FORMULA

Product_{n>=1} (1 + x^n)^a(n) = Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).

EXAMPLE

(1 + x) * (1 + x^2)^3 * (1 + x^3)^3 * (1 + x^4)^8 * (1 + x^5)^7 * ... * (1 + x^n)^a(n) * ... = 1/((1 - x) * (1 - x^2)^2 * (1 - x^3)^3 * (1 - x^4)^5 * (1 - x^5)^7 * ... * (1 - x^k)^p(k) * ...).

MATHEMATICA

nn = 40; f[x_] := Product[(1 + x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Product[1/(1 - x^k)^PartitionsP[k], {k, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

CROSSREFS

Cf. A000041, A001511, A001970.

Sequence in context: A266560 A021751 A302675 * A331530 A199624 A093366

Adjacent sequences:  A305838 A305839 A305840 * A305842 A305843 A305844

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jun 11 2018

STATUS

approved

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Last modified March 30 12:10 EDT 2020. Contains 333125 sequences. (Running on oeis4.)