A305841 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001970 (partitions of partitions).

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

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 A363220

Adjacent sequences: A305838 A305839 A305840 * A305842 A305843 A305844

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 11 2018

Status

approved