%I #22 Jan 10 2017 14:08:12

%S 1,2,4,8,12,20,32,48,72,106,152,216,304,420,576,784,1056,1412,1876,

%T 2472,3240,4224,5472,7056,9056,11566,14712,18640,23520,29572,37056,

%U 46272,57600,71488,88456,109152,134332,164884,201888,246608,300528,365428,443392,536856

%N G.f. = theta_4(0,x^4)/theta_4(0,x).

%C G.f. for the number of partitions of 2n in which all odd parts occur with multiplicities 2, 4 or 6. The even parts appear at most three times. E.g., a(8)=12 because 8 = 6+2 = 6+1+1 = 4+4 = 4+2+2 = 4+2+1+1 = 4+1+1+1+1 = 3+3+2 = 3+3+1+1 = 2+2+2+1+1 = 2+2+1+1+1+1 = 2+1+1+1+1+1+1.

%C Also the number of partitions of 2n in which the even parts appear with 2 types c, c* and with multiplicity 1. The odd parts with multiplicity 4. E.g., a(6)=8 because we have 6, 6*, 42, 42*, 4*2, 4*2*, 21111, 2*1111.

%H Vaclav Kotesovec, <a href="/A103258/b103258.txt">Table of n, a(n) for n = 0..1000</a>

%H Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.

%F Expansion of eta(q^2)*eta(q^4)^2/(eta(q)^2*eta(q^8)) in powers of q.

%F Euler transform of period 8 sequence [2, 1, 2, -1, 2, 1, 2, 0, ...]. - _Michael Somos_, Feb 10 2005

%F G.f.: product_{k>0} ((1+x^k)^2 * (1+x^(2*(2*k-1)))).

%F From _Vaclav Kotesovec_, Jan 10 2017: (Start)

%F a(n) ~ sqrt(3)*Pi * BesselI(1, sqrt(3*n)*Pi/2) / (8*sqrt(n)).

%F a(n) ~ 3^(1/4) * exp(Pi*sqrt(3*n)/2) / (8*n^(3/4)) * (1 - sqrt(3)/(4*Pi*sqrt(n)) - 5/(32*Pi^2*n)).

%F (End)

%t nmax = 60; CoefficientList[Series[Product[(1-x^(4*k)) * (1-x^(8*k-4)) * (1+x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%o (PARI) {a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^4+A)^2/eta(x+A)^2/eta(x^8+A), n))} /* _Michael Somos_, Feb 10 2005 */

%Y Cf. A002448.

%K nonn

%O 0,2

%A _Noureddine Chair_, Jan 27 2005