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A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x.

%I A274327 #53 Feb 16 2025 08:33:36
%S A274327 1,4,14,40,104,248,560,1200,2474,4924,9520,17928,33008,59528,105408,
%T A274327 183536,314744,532208,888382,1465208,2389808,3857456,6166096,9766576,
%U A274327 15336816,23888844,36924656,56659296,86341664,130710104,196640576,294059872,437232746,646561792
%N A274327 Expansion of Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4 in powers of x.
%H A274327 Seiichi Manyama, <a href="/A274327/b274327.txt">Table of n, a(n) for n = 0..10000</a>
%H A274327 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%F A274327 G.f.: Product_{n>=1} (1 - x^(4*n))/(1 - x^n)^4.
%F A274327 a(n) ~ 5*exp(Pi*sqrt(5*n/2)) / (2^(13/2) * n^(3/2)). - _Vaclav Kotesovec_, Nov 10 2016
%F A274327 G.f.: (x^4; x^4)_inf/((x; x)_inf)^4, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Nov 20 2016
%F A274327 a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A285895(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 29 2017
%e A274327 G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 104*x^4 + 248*x^5 + 560*x^6 + ...
%t A274327 nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 10 2016 *)
%t A274327 (QPochhammer[x^4, x^4]/QPochhammer[x, x]^4 + O[x]^40)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2016 *)
%o A274327 (PARI) first(n)=my(x='x);Vec(prod(k=1,n,(1-x^(4*k))/(1-x^k)^4,1+O(x^(n+1)))) \\ _Charles R Greathouse IV_, Nov 07 2016
%Y A274327 Cf. Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), A273845 (k=3), this sequence (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).
%Y A274327 Cf. A083703.
%K A274327 nonn
%O A274327 0,2
%A A274327 _Seiichi Manyama_, Nov 07 2016