A101230 Number of partitions of 2n in which both odd parts and parts that are multiples of 3 occur with even multiplicities. There is no restriction on the other even parts.
1, 2, 4, 7, 12, 20, 32, 50, 76, 113, 166, 240, 343, 484, 676, 935, 1282, 1744, 2355, 3158, 4208, 5573, 7340, 9616, 12536, 16266, 21012, 27028, 34628, 44196, 56204, 71226, 89964, 113270, 142180, 177948, 222089, 276430, 343172, 424959, 524966
Offset: 0
Keywords
Examples
a(8)=12 because 8 = 4+4 = 4+2+2 = 4+2+1+1 = 4+1+1+1+1 = 3+3+2 = 3+3+1+1 = 2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1. 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + 113*x^9 + ... 1/q + 2*q^7 + 4*q^15 + 7*q^23 + 12*q^31 + 20*q^39 + 32*q^47 + 50*q^55 + 76*q^63 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- 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^k)*(1+x^(3*k))),k=1..100),x=0,100);
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Mathematica
nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1))*(1+x^(3*k-2)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *) -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))} /* Michael Somos, Mar 04 2012 */
Formula
G.f.: product_{k>0}(1+x^k)/((1-x^k)(1+x^(3k)))= Theta_4(0, x^3)/theta(0, x)1/product_{k>0}(1-x^(3k)).
Euler transform of period 6 sequence [2, 1, 1, 1, 2, 1, ...]. - Vladeta Jovovic, Dec 17 2004
Expansion of q^(1/8) * eta(q^2) * eta(q^3) / (eta(q)^2 * eta(q^6)) in powers of q. - Michael Somos, Mar 04 2012
Convolution inverse of A089812. - Michael Somos, Mar 04 2012
a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*n). - Vaclav Kotesovec, Sep 01 2015
Comments