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

0,3

Links

Table of n, a(n) for n=0..32.

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).

Prog

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

Crossrefs

Cf. A027998, A052812, A255835, A363600.

Cf. A363601.

Sequence in context: A066143 A110045 A108931 * A307397 A032100 A340729

Adjacent sequences: A363596 A363597 A363598 * A363600 A363601 A363602

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 10 2023

Extensions

Name suggested by Joerg Arndt, Jun 11 2023

Status

approved