A363599 Number of partitions of n into distinct parts where there are k^2-1 kinds of part k.
1, 0, 3, 8, 18, 48, 109, 264, 594, 1360, 2988, 6552, 14115, 30048, 63288, 131800, 271953, 555792, 1126583, 2264472, 4518051, 8948544, 17603781, 34405272, 66828247, 129040704, 247765665, 473160696, 898924929, 1699331808, 3197083220, 5987288352, 11162934948
Offset: 0
Programs
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PARI
my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2-1)))
Formula
G.f.: Product_{k>=1} (1+x^k)^(k^2-1).
a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2-1) ) * a(n-k).
Extensions
Name suggested by Joerg Arndt, Jun 11 2023