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
Keywords
Examples
(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) * ...).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..2000
- N. J. A. Sloane, Transforms
Programs
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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
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).
Comments