A026833 Number of partitions of n into distinct parts, the least being even.

0, 0, 1, 0, 1, 1, 2, 1, 2, 3, 4, 4, 5, 6, 8, 9, 11, 14, 16, 18, 22, 26, 31, 36, 42, 49, 57, 66, 76, 88, 102, 116, 134, 154, 176, 201, 229, 260, 296, 336, 381, 432, 488, 550, 622, 700, 788, 886, 994, 1115, 1250, 1399, 1564, 1748, 1952, 2176, 2426, 2701, 3004

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Also number of partitions of n such that if k is the largest part, then k occurs an even number of times and each of the numbers 1,2,...,k-1 occurs at least once. Example: a(10)=4 because we have [3,3,2,1,1], [2,2,2,2,1,1], [2,2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 30 2006

Examples

			a(10)=4 because we have [10], [8,2], [6,4] and [5,3,2].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A026832.

Programs

Maple

g:=sum(x^(2*k)*product(1+x^j, j=2*k+1..60), k=1..60): gser:=series(g, x=0, 58): seq(coeff(gser, x, n), n=0..55); # Emeric Deutsch, Mar 30 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2-1 b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2019

Mathematica

b[n_, i_] := b[n, i] = If[i*(i+1)/2-1 < n, 0, b[n, i-1] + If[i == n && EvenQ[i], 1, 0]+If[i < n, b[n-i, Min[n-i, i-1]], 0]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 16 2025, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=2} ((-1)^k*(-1+Product_{i>=k} (1+x^i))). - Vladeta Jovovic, Aug 26 2003

G.f.: Sum_{k>=1} x^(2k)*Product_{j>=2k+1} (1+x^j).

G.f.: Sum_{k>=1} x^(k*(k+3)/2)/((1+x^k)*Product_{j=1..k} (1-x^j)). - Emeric Deutsch, Mar 30 2006

a(n) ~ exp(Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 09 2019

Extensions

a(0)=0 prepended by Alois P. Heinz, Feb 01 2019