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A282942 Expansion of Product_{k>=1} (1 - q^k)^8/(1 - q^(7*k)) in powers of q.

%I A282942 #29 Sep 08 2022 08:46:18
%S A282942 1,-8,20,0,-70,64,56,1,-133,-140,308,-70,174,56,-518,-141,-63,868,
%T A282942 -140,238,294,-1029,-1154,-203,2366,-658,1296,350,-1547,-1295,-1666,
%U A282942 3234,-2128,2534,2464,-2577,-3087,-609,5600,-2716,2435,294,-3745,-4249,-1015,8526
%N A282942 Expansion of Product_{k>=1} (1 - q^k)^8/(1 - q^(7*k)) in powers of q.
%D A282942 G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 192.
%H A282942 Seiichi Manyama, <a href="/A282942/b282942.txt">Table of n, a(n) for n = 0..1000</a>
%F A282942 G.f.: exp( Sum_{n>=1} -sigma(7*n)*q^n/n ). - _Seiichi Manyama_, Mar 04 2017
%F A282942 a(n) = -(1/n)*Sum_{k=1..n} sigma(7*k)*a(n-k). - _Seiichi Manyama_, Mar 04 2017
%e A282942 G.f.: 1 - 8*q + 20*q^2 - 70*q^4 + 64*q^5 + 56*q^6 + q^7 - 133*q^8 - 140*q^9 + ...
%t A282942 nmax = 50; CoefficientList[Series[Product[(1-x^k)^8/(1-x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _G. C. Greubel_, Nov 18 2018 *)
%o A282942 (PARI) m=50; x='x+O('x^m); Vec(prod(j=1,m, (1-x^j)^8/(1-x^(7*j)))) \\ _G. C. Greubel_, Nov 18 2018
%o A282942 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^j)^8/(1-x^(7*j)): j in [1..m]]) )); // _G. C. Greubel_, Nov 18 2018
%o A282942 (Sage)
%o A282942 R = PowerSeriesRing(ZZ, 'x')
%o A282942 x = R.gen().O(50)
%o A282942 s = prod((1-x^j)^8/(1-x^(7*j)) for j in (1..50))
%o A282942 list(s) # _G. C. Greubel_, Nov 18 2018
%Y A282942 Cf. A282941.
%Y A282942 Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), A282937 (k=5), this sequence (k=7).
%K A282942 sign
%O A282942 0,2
%A A282942 _Seiichi Manyama_, Feb 25 2017