A103257 Number of partitions of 2n free of multiples of 5. All odd parts occur with multiplicity 2 or 4. the even parts occur at most twice.

1, 2, 4, 6, 10, 14, 20, 28, 40, 54, 72, 96, 126, 164, 212, 272, 346, 436, 548, 684, 850, 1052, 1296, 1588, 1940, 2362, 2864, 3462, 4172, 5012, 6004, 7172, 8548, 10160, 12048, 14256, 16830, 19828, 23312, 27356, 32040

Offset: 0

Author

Noureddine Chair, Jan 27 2005

Keywords

• nonn

Comments

Convolution of A261796 and A261797. - Vaclav Kotesovec, Sep 01 2015

Examples

			E.g. a(5) = 14 because 10 can be written as 8+2 = 8+1+1 = 6+4 = 6+2+2 = 6+2+1+1 = 6+1+1+1+1 = 4+4+2 = 4+4+1+1 = 4+3+3 = 4+2+2+1+1 = 4+2+1+1+1+1 = 3+3+2+2 = 3+3+2+1+1 = 3+3+1+1+1+1.
		

Links

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

Crossrefs

Cf. A098151, A261796, A261797.

Programs

Maple

series(product(((1+x^k)*(1-x^(3*k))*(1-x^(5*k)))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))),k=1..100),x=0,100);

Mathematica

nmax = 50; CoefficientList[Series[Product[((1+x^k)*(1-x^(3*k))*(1-x^(5*k)))/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

PARI

q='q+O('q^33); E(k)=eta(q^k);
Vec( (E(2)*E(3)^2*E(5)^2) / (E(1)^2*E(6)*E(10)) ) \\ Joerg Arndt, Sep 01 2015

Formula

G.f.: (theta_4(0, x^3)*theta_4(0, x^5))/theta_4(0, x).

G.f.: (E(2)*E(3)^2*E(5)^2) / (E(1)^2*E(6)*E(10)) where E(k) = Product_{n>=1} 1-q^(k*n). - Joerg Arndt, Sep 01 2015

a(n) ~ exp(Pi*sqrt(7*n/15)) / sqrt(15*n). - Vaclav Kotesovec, Sep 01 2015

Extensions

Example corrected by Vaclav Kotesovec, Sep 01 2015

Maple program fixed by Vaclav Kotesovec, Sep 01 2015