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Number of partitions of n into distinct parts where there are k^2-1 kinds of part k.
1

%I #14 Jun 11 2023 11:14:50

%S 1,0,3,8,18,48,109,264,594,1360,2988,6552,14115,30048,63288,131800,

%T 271953,555792,1126583,2264472,4518051,8948544,17603781,34405272,

%U 66828247,129040704,247765665,473160696,898924929,1699331808,3197083220,5987288352,11162934948

%N Number of partitions of n into distinct parts where there are k^2-1 kinds of part k.

%F G.f.: Product_{k>=1} (1+x^k)^(k^2-1).

%F a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2-1) ) * a(n-k).

%o (PARI) my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2-1)))

%Y Cf. A027998, A052812, A255835, A363600.

%Y Cf. A363601.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jun 10 2023

%E Name suggested by _Joerg Arndt_, Jun 11 2023