A238589 Number of partitions p of n such that 2*min(p) is a part of p.

0, 0, 1, 1, 2, 4, 5, 8, 13, 17, 24, 36, 47, 64, 88, 116, 153, 203, 261, 340, 439, 559, 710, 905, 1136, 1427, 1786, 2223, 2756, 3415, 4201, 5167, 6330, 7730, 9413, 11449, 13864, 16767, 20225, 24344, 29228, 35045, 41898, 50029, 59609, 70899, 84165, 99785, 118052

Offset: 1

Author

Clark Kimberling, Mar 01 2014

Keywords

• nonn
• easy

Examples

			a(6) counts these partitions:  42, 321, 2211, 21111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A117989, A238590, A238591.

Cf. A118096, A238479, A238594.

Programs

Mathematica

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 2*Min[p]]], {n, 50}]

PARI

my(N=50, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/prod(j=k, N, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

Formula

a(n) = A000041(n) - A238594(n).

G.f.: Sum_{k>=1} x^(3*k)/Product_{j>=k} (1-x^j). - Seiichi Manyama, May 17 2023

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 - (sqrt(3/2)/Pi + 49*Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Jun 19 2025