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Number of partitions of 2n free of multiples of 8 such that 4 occurs at most once. All odd parts occur with even multiplicities. There is no restriction on the other even parts.
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%I #17 Aug 15 2016 18:16:25

%S 1,2,4,8,12,20,32,48,72,106,152,216,305,422,580,792,1068,1432,1908,

%T 2520,3313,4332,5628,7280,9373,12008,15324,19480,24661,31112,39120,

%U 49016,61229,76260,94692,117264,144834,178412,219244,268784,328746

%N Number of partitions of 2n free of multiples of 8 such that 4 occurs at most once. All odd parts occur with even multiplicities. There is no restriction on the other even parts.

%H Vaclav Kotesovec, <a href="/A100684/b100684.txt">Table of n, a(n) for n = 0..10000</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.: (1-x^4)*Product((1+x^(2*i))/(1-x^(2*i-1))^2, i=1..infinity). [_Vladeta Jovovic_]

%F Expansion of (1 - q^4) * q^(-1/6) * eta(q^4) * eta(q^2) / eta(q)^2 in powers of q.

%F G.f.: (1-x^4) * Prod_{k>0} (1 + x^(2*k)) * (1 + x^k)^2. - _Michael Somos_, Feb 10 2005

%F a(n) ~ 5^(3/4) * Pi * exp(Pi*sqrt(5*n/6)) / (2^(11/4) * 3^(3/4) * n^(5/4)). - _Vaclav Kotesovec_, Sep 06 2015

%e G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 48*x^7 + 72*x^8 + ...

%t nmax = 40; CoefficientList[Series[(1-x^4)*Product[(1+x^(2*k))/(1-x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 06 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( (1 - x^4) * eta(x^4 + A) * eta(x^2 + A) / eta(x + A)^2, n))}; /* _Michael Somos_, Feb 10 2005 */

%Y Cf. A080054.

%K nonn

%O 0,2

%A _Noureddine Chair_, Jan 27 2005

%E Corrected by _Vladeta Jovovic_, Feb 01 2005

%E Typo in PARI program fixed by _Vaclav Kotesovec_, Sep 06 2015