A100684 Number of partitions of 2n free of multiples of 8 such that 4 occurs at most once. All odd parts occur with even multiplicities. There is no restriction on the other even parts.

1, 2, 4, 8, 12, 20, 32, 48, 72, 106, 152, 216, 305, 422, 580, 792, 1068, 1432, 1908, 2520, 3313, 4332, 5628, 7280, 9373, 12008, 15324, 19480, 24661, 31112, 39120, 49016, 61229, 76260, 94692, 117264, 144834, 178412, 219244, 268784, 328746

Offset: 0

Author

Noureddine Chair, Jan 27 2005

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 48*x^7 + 72*x^8 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Crossrefs

Cf. A080054.

Programs

Mathematica

nmax = 40; CoefficientList[Series[(1-x^4)*Product[(1+x^(2*k))/(1-x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 06 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( (1 - x^4) * eta(x^4 + A) * eta(x^2 + A) / eta(x + A)^2, n))}; /* Michael Somos, Feb 10 2005 */

Formula

G.f.: (1-x^4)*Product((1+x^(2*i))/(1-x^(2*i-1))^2, i=1..infinity). [Vladeta Jovovic]

Expansion of (1 - q^4) * q^(-1/6) * eta(q^4) * eta(q^2) / eta(q)^2 in powers of q.

G.f.: (1-x^4) * Prod_{k>0} (1 + x^(2*k)) * (1 + x^k)^2. - Michael Somos, Feb 10 2005

a(n) ~ 5^(3/4) * Pi * exp(Pi*sqrt(5*n/6)) / (2^(11/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Sep 06 2015

Extensions

Corrected by Vladeta Jovovic, Feb 01 2005

Typo in PARI program fixed by Vaclav Kotesovec, Sep 06 2015