A035362 Number of partitions of n into parts 4k or 4k+1.

1, 1, 1, 2, 3, 3, 3, 5, 7, 8, 8, 11, 15, 17, 18, 23, 30, 35, 37, 45, 57, 66, 71, 84, 104, 121, 131, 151, 183, 212, 231, 263, 313, 362, 396, 446, 523, 601, 660, 738, 855, 979, 1076, 1196, 1372, 1562, 1719, 1903, 2164, 2454, 2701, 2979, 3363, 3795, 4177, 4594

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Comments

Also number of partitions of n such that number of 1's plus number of odd parts is greater than or equal to n. - Vladeta Jovovic, Feb 27 2006

Examples

			a(8)=5 because we have [8],[5,1,1,1],[4,4],[4,1,1,1,1] and [1,1,1,1,1,1,1,1].
		

Programs

Maple

g:=-1+1/(1-x)/product((1-x^(4*j))*(1-x^(4*j+1)),j=1..20): gser:=series(g,x=0,60): seq(coeff(gser,x^n),n=1..56); # Emeric Deutsch, Mar 07 2006

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(4k+4))*(1 - x^(4k+1))), {k, 0, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 27 2015 *)

Formula

G.f.: -1 + 1/((1-x)*Product_{j>=1} (1-x^(4j))*(1-x^(4j+1))). - Emeric Deutsch, Mar 07 2006

a(n) ~ exp(Pi*sqrt(n/3)) * Gamma(5/4) / (2^(1/4) * 3^(3/8) * Pi^(3/4) * n^(7/8)). - Vaclav Kotesovec, Aug 27 2015