A282931 - OEIS

%I #31 Sep 08 2022 08:46:18

%S 1,53,1484,29097,447426,5734918,63638001,627260142,5594403499,

%T 45779730871,347453597091,2466970932027,16501339314082,

%U 104588498225862,631215364345159,3642533720923593,20170341090888205,107511123136305075,553099301324196585

%N Expansion of Product_{k>=1} (1 - x^(7*k))^52/(1 - x^k)^53 in powers of x.

%H Seiichi Manyama, <a href="/A282931/b282931.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{n>=1} (1 - x^(7*n))^52/(1 - x^n)^53.

%F a(n) ~ exp(Pi*sqrt(638*n/21)) * sqrt(319) / (4*sqrt(3) * 7^(53/2) * n). - _Vaclav Kotesovec_, Nov 10 2017

%t nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^52/(1 - x^k)^53, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 10 2017 *)

%o (PARI) my(N=30,x='x+O('x^N)); Vec(prod(j=1,N, (1 - x^(7*j))^52/(1 - x^j)^53)) \\ _G. C. Greubel_, Nov 18 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^52/(1 - x^j)^53: j in [1..30]]) )); // _G. C. Greubel_, Nov 18 2018

%o (Sage)

%o R = PowerSeriesRing(ZZ, 'x')

%o prec = 30

%o x = R.gen().O(prec)

%o s = prod((1 - x^(7*j))^52/(1 - x^j)^53 for j in (1..prec))

%o print(s.coefficients()) # _G. C. Greubel_, Nov 18 2018

%Y Cf. A282919.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 24 2017