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

0,2

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

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) = prod(n>=1, 1-q^k ). - Joerg Arndt, Sep 01 2015

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A098151, A261796, A261797.

Sequence in context: A268752 A277277 A241337 * A103259 A280131 A082380

Adjacent sequences: A103254 A103255 A103256 * A103258 A103259 A103260

Keyword

nonn

Author

Noureddine Chair, Jan 27 2005

Extensions

Example corrected by Vaclav Kotesovec, Sep 01 2015

Maple program fixed by Vaclav Kotesovec, Sep 01 2015

Status

approved