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Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001970 (partitions of partitions).
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%I #8 Jun 11 2018 15:31:54

%S 1,3,3,8,7,14,15,30,30,49,56,91,101,150,176,261,297,415,490,676,792,

%T 1058,1255,1666,1958,2537,3010,3868,4565,5780,6842,8610,10143,12607,

%U 14883,18392,21637,26505,31185,38014,44583,53966,63261,76233,89134,106813,124754

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

%C Inverse weigh transform of A001970.

%H Alois P. Heinz, <a href="/A305841/b305841.txt">Table of n, a(n) for n = 1..2000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F 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).

%e (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) * ...).

%t 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

%Y Cf. A000041, A001511, A001970.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Jun 11 2018