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

1,2

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

Links

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

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A238589, A325535.

Sequence in context: A195012 A333192 A323092 * A350837 A239945 A363740

Adjacent sequences: A238591 A238592 A238593 * A238595 A238596 A238597

Keyword

nonn,easy

Author

Clark Kimberling, Mar 01 2014

Status

approved