A238594 Number of partitions p of n such that 2*min(p) is not a part of p.

1, 2, 2, 4, 5, 7, 10, 14, 17, 25, 32, 41, 54, 71, 88, 115, 144, 182, 229, 287, 353, 443, 545, 670, 822, 1009, 1224, 1495, 1809, 2189, 2641, 3182, 3813, 4580, 5470, 6528, 7773, 9248, 10960, 12994, 15355, 18129, 21363, 25146, 29525, 34659, 40589, 47488, 55473

Offset: 1

Author

Clark Kimberling, Mar 01 2014

Keywords

• nonn
• easy

Comments

a(n) is also the number of partitions of n with a part whose multiplicity is greater than half the total number of parts. - Andrew Howroyd, Jan 17 2024

Examples

			a(6) counts all 11 partitions of 6 except these: 42, 321, 2211, 21111.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (using data from A238589)

Crossrefs

Cf. A238589, A325535.

Programs

Mathematica

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

PARI

seq(n) = {Vec(sum(k=1, n\2+1, x^(2*k-2)*(1 + x - x^(k-1))/prod(j=1, k, 1 - x^j, 1 + O(x^(n-2*k+3))), O(x*x^n)))} \\ Andrew Howroyd, Jan 17 2024

Formula

a(n) = A000041(n) - A238589(n).

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*2^(3/2)*n^(3/2)). - Vaclav Kotesovec, Jun 09 2021

a(n) = Sum_{k>=1} x^(2*k-2)*(1 + x - x^(k-1))/(Product_{j=1..k} (1 - x^j)). - Andrew Howroyd, Jan 17 2024