A096778 Number of partitions of n with at most two even parts.

1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 34, 45, 58, 75, 95, 121, 151, 189, 234, 289, 354, 433, 526, 637, 768, 923, 1105, 1319, 1569, 1861, 2202, 2597, 3056, 3587, 4201, 4908, 5723, 6658, 7732, 8961, 10367, 11971, 13802, 15884, 18253, 20942, 23992, 27445, 31353

Offset: 0

Author

Vladeta Jovovic, Aug 16 2004

Keywords

• easy
• nonn

Comments

Also number of partitions of n+4 with exactly two even parts. Example: a(3)=3 because the partitions of 7 with exactly two even parts are [4,2,1], [3,2,2] and [2,2,1,1,1]. a(n)=A116482(n+4,2). - Emeric Deutsch, Feb 21 2006

Examples

			a(3)=3 because we have [3],[2,1] and [1,1,1].
		

References

• Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=2. - N. J. A. Sloane, Aug 31 2014

Crossrefs

Cf. A038348.

Cf. A116482.

Programs

Mathematica

CoefficientList[ Series[(1/((1 - x^2)*(1 - x^4)))/Product[1 - x^(2i + 1), {i, 0, 50}], {x, 0, 48}], x] (* Robert G. Wilson v, Aug 16 2004 *)

Formula

G.f.: (1/((1-x^2)*(1-x^4)))/Product(1-x^(2*i+1), i=0..infinity). More generally, g.f. for number of partitions of n with at most k even parts is (1/Product(1-x^(2*i), i=1..k))/Product(1-x^(2*i+1), i=0..infinity).

a(n) ~ 3^(3/4) * n^(1/4) * exp(Pi*sqrt(n/3)) / (8*Pi^2). - Vaclav Kotesovec, May 29 2018

Extensions

More terms from Robert G. Wilson v, Aug 17 2004

More terms from Emeric Deutsch, Feb 21 2006