A363214 Number of partitions of n with rank 5 (the rank of a partition is the largest part minus the number of parts).

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 7, 11, 13, 19, 21, 30, 35, 47, 55, 73, 85, 111, 131, 166, 197, 247, 291, 362, 428, 525, 620, 756, 890, 1078, 1268, 1523, 1791, 2140, 2507, 2983, 3490, 4131, 4824, 5688, 6626, 7785, 9052, 10595, 12298, 14351, 16618, 19339, 22355, 25938

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=5 in the triangle A063995.

Column r=5 of A105806.

Cf. A000041.

Programs

Mathematica

Table[Count[IntegerPartitions[n],?(#[[1]]-Length[#]==5&)],{n,60}] (* _Harvey P. Dale, Feb 24 2024 *)

PARI

my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(5*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^(5*k) * ( x^(k*(3*k-1)/2) - x^(k*(3*k+1)/2) ).