A363213 Number of partitions of n with rank 4 (the rank of a partition is the largest part minus the number of parts).

0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 11, 12, 18, 20, 28, 33, 44, 51, 68, 80, 103, 122, 154, 182, 229, 270, 334, 396, 485, 572, 698, 822, 993, 1169, 1404, 1649, 1971, 2310, 2745, 3214, 3803, 4439, 5235, 6099, 7162, 8331, 9750, 11315, 13205, 15294, 17794, 20574, 23872

Offset: 1

Author

Seiichi Manyama, May 21 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Column k=4 in the triangle A063995.

Column r=4 of A105806.

Cf. A000041.

Programs

PARI

my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(4*k)*(x^(k*(3*k-1)/2)-x^(k*(3*k+1)/2)))))

Formula

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(4*k) * ( x^(k*(3*k-1)/2) - x^(k*(3*k+1)/2) ).