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%I #22 Apr 14 2017 23:54:54
%S 1,2,4,7,12,20,32,50,76,113,166,240,343,484,676,935,1282,1744,2355,
%T 3158,4208,5573,7340,9616,12536,16266,21012,27028,34628,44196,56204,
%U 71226,89964,113270,142180,177948,222089,276430,343172,424959,524966
%N 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.
%C Note that if a partition of n has odd parts occur with even multiplicities then n must be even. This is the reason for only looking at partitions of 2n. - _Michael Somos_, Mar 04 2012
%H G. C. Greubel, <a href="/A101230/b101230.txt">Table of n, a(n) for n = 0..1000</a>
%H Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.
%F 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)).
%F Euler transform of period 6 sequence [2, 1, 1, 1, 2, 1, ...]. - _Vladeta Jovovic_, Dec 17 2004
%F 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
%F Convolution inverse of A089812. - _Michael Somos_, Mar 04 2012
%F Convolution product of A000041 and A003105. - _Michael Somos_, Mar 04 2012
%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*n). - _Vaclav Kotesovec_, Sep 01 2015
%e 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.
%e 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 + ...
%e 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 + ...
%p series(product((1+x^k)/((1-x^k)*(1+x^(3*k))),k=1..100),x=0,100);
%t 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 *)
%o (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 */
%Y Cf. A000041, A003105, A015128, A089812, A098151.
%K nonn
%O 0,2
%A _Noureddine Chair_, Dec 16 2004
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