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Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^4.
2

%I #4 Aug 28 2015 03:16:04

%S 1,4,10,20,35,60,100,160,245,364,536,780,1115,1564,2166,2980,4065,

%T 5484,7326,9720,12830,16824,21902,28344,36510,46820,59736,75844,95910,

%U 120844,151688,189668,236330,293564,363542,448804,552425,678144,830338,1014052,1235296

%N Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^4.

%C In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} 1/(1 - x^(a*k+b))^j, then a(n) ~ Gamma(b/a)^j * 2^(-(j+5)/4 - j*b/(2*a)) * 3^((j-1)/4 - j*b/(2*a)) * j^(-(j-1)/4 + j*b/(2*a)) * a^(-(j+1)/4 + j*b/(2*a)) * Pi^(-j + j*b/a) * n^((j-3)/4 - j*b/(2*a)) * exp(Pi*sqrt(2*j*n/(3*a))).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(1/4) * Gamma(1/4)^4 / (32 * Pi^3 * n^(1/4)).

%t nmax=50; CoefficientList[Series[Product[1/(1-x^(4*k+1))^4, {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A035382, A261616, A261635, A035451, A261629, A261632.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 27 2015