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A277974 Expansion of ((Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^5) - 1)/5 in powers of x. 7

%I

%S 0,1,4,13,38,101,252,594,1340,2907,6104,12447,24744,48068,91476,

%T 170838,313646,566824,1009628,1774290,3079338,5282172,8962288,

%U 15050848,25032420,41255101,67406472,109236685,175654072,280371510,444372452,699579062,1094289564

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

%H Seiichi Manyama, <a href="/A277974/b277974.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A277212(n)/5, n > 0.

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

%F a(n) ~ exp(4*Pi*sqrt(n/5)) / (sqrt(2) * 5^(11/4) * n^(7/4)). - _Vaclav Kotesovec_, Nov 10 2016

%e G.f. = x + 4*x^2 + 13*x^3 + 38*x^4 + 101*x^5 + 252*x^6 + ...

%t nmax = 50; CoefficientList[Series[(Product[(1 - x^(5*j))/(1 - x^j)^5, {j, 1, nmax}] - 1)/5, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 10 2016 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5] / QPochhammer[ x]^5 - 1) / 5, {x, 0, n}]; (* _Michael Somos_, Nov 13 2016 *)

%o (PARI) x='x+O('x^66); concat([0],Vec(eta(x^5)/eta(x)^5-1)/5) \\ _Joerg Arndt_, Nov 27 2016

%Y Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), A277968 (k=3), this sequence (k=5), A160549 (k=7), A277912 (k=11).

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 07 2016

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Last modified June 17 04:26 EDT 2019. Contains 324183 sequences. (Running on oeis4.)