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Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4 in powers of x.
2

%I #18 Nov 10 2017 05:19:25

%S 1,4,14,40,105,251,570,1226,2540,5075,9855,18630,34439,62340,110805,

%T 193624,333235,565415,947040,1567130,2564425,4152535,6658711,10579380,

%U 16663755,26033200,40357641,62106290,94912385,144088840,217368655,325945320,485950150,720515475

%N Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4 in powers of x.

%H Seiichi Manyama, <a href="/A278680/b278680.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4.

%F a(n) ~ 19 * exp(Pi*sqrt(38*n/15)) / (120 * sqrt(10) * n^(3/2)). - _Vaclav Kotesovec_, Nov 10 2017

%e G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 105*x^4 + 251*x^5 + 570*x^6 + ...

%t nmax = 30; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 10 2017 *)

%Y Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), A278668 (k=3), this sequence (k=4), A277212 (k=5), A182821 (k=6).

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 25 2016