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A282929 Expansion of Product_{k>=1} (1 - x^(7*k))^44/(1 - x^k)^45 in powers of x.

%I A282929 #35 Sep 08 2022 08:46:18
%S A282929 1,45,1080,18285,244260,2733804,26606745,230915656,1819708110,
%T A282929 13198528010,89041203249,563420646090,3366705675744,19105222953420,
%U A282929 103448715353372,536621238174195,2675953974595655,12866398610335149,59805282183021050,269356649381129943,1177903345233332970,5010462608512204473,20765528801742226455
%N A282929 Expansion of Product_{k>=1} (1 - x^(7*k))^44/(1 - x^k)^45 in powers of x.
%H A282929 Seiichi Manyama, <a href="/A282929/b282929.txt">Table of n, a(n) for n = 0..1000</a>
%F A282929 G.f.: Product_{n>=1} (1 - x^(7*n))^44/(1 - x^n)^45.
%F A282929 a(n) ~ exp(Pi*sqrt(542*n/21)) * sqrt(271) / (4*sqrt(3) * 7^(45/2) * n). - _Vaclav Kotesovec_, Nov 10 2017
%p A282929 N:= 30:
%p A282929 gN:= mul((1-x^(7*n))^44/(1-x^n)^45,n=1..N):
%p A282929 S:=series(gN,x,N+1):
%p A282929 seq(coeff(S,x,n),n=1..N); # _Robert Israel_, Nov 18 2018
%t A282929 nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^44/(1 - x^k)^45, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 10 2017 *)
%o A282929 (PARI) my(N=30,x='x+O('x^N)); Vec(prod(j=1, N, (1 - x^(7*j))^44/(1 - x^j)^45)) \\ _G. C. Greubel_, Nov 18 2018
%o A282929 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^44/(1 - x^j)^45: j in [1..m+2]]) )); // _G. C. Greubel_, Nov 18 2018
%o A282929 (Sage)
%o A282929 R = PowerSeriesRing(ZZ, 'x')
%o A282929 prec = 30
%o A282929 x = R.gen().O(prec)
%o A282929 s = prod((1 - x^(7*j))^44/(1 - x^j)^45 for j in (1..prec))
%o A282929 print(s.coefficients()) # _G. C. Greubel_, Nov 18 2018
%Y A282929 Cf. A282919.
%K A282929 nonn
%O A282929 0,2
%A A282929 _Seiichi Manyama_, Feb 24 2017