A060285 Number of partitions of n objects of 2 colors with parts size >1.

1, 0, 3, 4, 11, 18, 42, 70, 144, 248, 466, 802, 1442, 2444, 4247, 7116, 12030, 19878, 32938, 53670, 87429, 140680, 225815, 359100, 569157, 895224, 1402941, 2184662, 3388915, 5228458, 8035921, 12291710, 18732318, 28425342, 42981877, 64740330

Offset: 0

Author

Vladeta Jovovic, Mar 23 2001

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vaclav Kotesovec)
• N. J. A. Sloane, Transforms

Crossrefs

Cf. (row sums of) A060244, A054225, A005380.

Programs

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k+1),{k,2,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 04 2015 *)

Formula

Euler transform of sequence [0, 3, 4, 5, 6, ...].

G.f.: Product_{k=2..infinity} 1/(1-x^k)^(k+1).

From Vaclav Kotesovec, Mar 09 2015: (Start)

For n>=2, a(n) = A005380(n-2) - 2*A005380(n-1) + A005380(n).

a(n) ~ 2^(1/36) * Zeta(3)^(37/36) * exp(1/12 - Pi^4/(432*Zeta(3)) + Pi^2 * n^(1/3) / (3*2^(4/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 3^(1/2) * Pi * n^(55/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .

a(n) ~ (2*Zeta(3))^(2/3) * A005380(n) / n^(2/3).

(End)

Extensions

Edited by Christian G. Bower, Jan 08 2004