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

1,10

Links

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

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) ).

Mathematica

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

Prog

(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)))))

Crossrefs

Column k=5 in the triangle A063995.

Column r=5 of A105806.

Cf. A000041.

Sequence in context: A304178 A266776 A371514 * A062896 A025065 A306664

Adjacent sequences: A363211 A363212 A363213 * A363215 A363216 A363217

Keyword

nonn

Author

Seiichi Manyama, May 21 2023

Status

approved