cp's OEIS Frontend

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

%I A305841 #8 Jun 11 2018 15:31:54
%S A305841 1,3,3,8,7,14,15,30,30,49,56,91,101,150,176,261,297,415,490,676,792,
%T A305841 1058,1255,1666,1958,2537,3010,3868,4565,5780,6842,8610,10143,12607,
%U A305841 14883,18392,21637,26505,31185,38014,44583,53966,63261,76233,89134,106813,124754
%N A305841 Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001970 (partitions of partitions).
%C A305841 Inverse weigh transform of A001970.
%H A305841 Alois P. Heinz, <a href="/A305841/b305841.txt">Table of n, a(n) for n = 1..2000</a>
%H A305841 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A305841 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 A305841 (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 A305841 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 A305841 Cf. A000041, A001511, A001970.
%K A305841 nonn
%O A305841 1,2
%A A305841 _Ilya Gutkovskiy_, Jun 11 2018