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
Keywords
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.
Programs
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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);
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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
Comments