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%I A282932 #34 Sep 08 2022 08:46:18
%S A282932 1,57,1710,35815,586815,7997157,94175267,983458849,9279004863,
%T A282932 80218101555,642408637594,4807304399931,33855173217278,
%U A282932 225702273908048,1431470152072364,8673471170235715,50389686887219910,281575909008910196,1517580284619183809
%N A282932 Expansion of Product_{k>=1} (1 - x^(7*k))^56/(1 - x^k)^57 in powers of x.
%C A282932 In general, if m >= 1 and g.f. = Product_{k>=1} (1 - x^(7*k))^m / (1 - x^k)^(m+1), then a(n) ~ exp(Pi*sqrt((2*(6*m+7)*n)/21)) * sqrt(6*m+7)) / (4*sqrt(3) * 7^((m+1)/2) * n). - _Vaclav Kotesovec_, Nov 10 2017
%H A282932 Seiichi Manyama, <a href="/A282932/b282932.txt">Table of n, a(n) for n = 0..1000</a>
%F A282932 G.f.: Product_{n>=1} (1 - x^(7*n))^56/(1 - x^n)^57.
%F A282932 a(n) ~ exp(Pi*sqrt(686*n/21)) * sqrt(343) / (4*sqrt(3) * 7^(57/2) * n). - _Vaclav Kotesovec_, Nov 10 2017
%t A282932 nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^56/(1 - x^k)^57, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 10 2017 *)
%o A282932 (PARI) my(N=30,x='x+O('x^N)); Vec(prod(j=1,N, (1 - x^(7*j))^56/(1 - x^j)^57)) \\ _G. C. Greubel_, Nov 18 2018
%o A282932 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^56/(1 - x^j)^57: j in [1..m+2]]) )); // _G. C. Greubel_, Nov 18 2018
%o A282932 (Sage)
%o A282932 R = PowerSeriesRing(ZZ, 'x')
%o A282932 prec = 30
%o A282932 x = R.gen().O(prec)
%o A282932 s = prod((1 - x^(7*j))^56/(1 - x^j)^57 for j in (1..prec))
%o A282932 print(s.coefficients()) # _G. C. Greubel_, Nov 18 2018
%Y A282932 Cf. A282919.
%K A282932 nonn
%O A282932 0,2
%A A282932 _Seiichi Manyama_, Feb 24 2017

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