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

0,7

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

Links

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

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

Example

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

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<n, 0, b(n, i-1)+
      `if`(i=n and i::even, 1, 0)+`if`(i<n, b(n-i, min(n-i, i-1)), 0))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 01 2019

Crossrefs

Cf. A026832.

Sequence in context: A031231 A030562 A238219 * A281544 A056882 A035534

Adjacent sequences: A026830 A026831 A026832 * A026834 A026835 A026836

Keyword

nonn

Author

Clark Kimberling

Extensions

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

Status

approved