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A285442 Expansion of Product_{k>0} ((1-x^{5k-2}) * (1-x^{5k-3})/((1-x^{5k-1}) * (1-x^{5k-4})))^2 in powers of x.

%I A285442 #28 Jul 29 2024 06:14:19
%S A285442 1,2,1,-2,-2,2,5,0,-8,-6,7,14,1,-18,-15,14,30,2,-40,-32,32,66,6,-82,
%T A285442 -65,60,125,8,-157,-120,117,238,19,-286,-222,206,419,28,-507,-386,366,
%U A285442 732,55,-864,-659,610,1224,86,-1442,-1090,1016,2024,147,-2350,-1775,1632
%N A285442 Expansion of Product_{k>0} ((1-x^{5k-2}) * (1-x^{5k-3})/((1-x^{5k-1}) * (1-x^{5k-4})))^2 in powers of x.
%H A285442 Seiichi Manyama, <a href="/A285442/b285442.txt">Table of n, a(n) for n = 0..10000</a>
%H A285442 R. P. Agarwal, <a href="https://www.ias.ac.in/article/fulltext/pmsc/103/03/0269-0293">Lambert series and Ramanujan</a>, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 282-283.
%F A285442 a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A109091(k)*a(n-k) for n > 0.
%F A285442 Expansion of square of continued fraction 1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...)))). - _Ilya Gutkovskiy_, Apr 19 2017
%F A285442 From _Seiichi Manyama_, Jul 29 2024: (Start)
%F A285442 G.f.: ( Sum_{k in Z} x^k / (1 - x^(5*k+1)) ) / ( Sum_{k in Z} x^(3*k) / (1 - x^(5*k+1)) ).
%F A285442 G.f.: ( Sum_{k in Z} x^k / (1 - x^(5*k+2)) ) / ( Sum_{k in Z} x^(2*k) / (1 - x^(5*k+2)) ). (End)
%t A285442 nmax = 60; CoefficientList[Series[Product[((1-x^(5k-2)) * (1-x^(5k-3)) / ((1-x^(5k-1)) * (1-x^(5k-4))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 13 2017 *)
%Y A285442 Product_{k>0} ((1-x^{5k-1}) * (1-x^{5k-4})/((1-x^{5k-2}) * (1-x^{5k-3})))^m: A285444 (m=-4), A285443 (m=-3), this sequence (m=-2), A003823 (m=-1), A007325 (m=1), A055101 (m=2), A055102 (m=3), A055103 (m=4).
%Y A285442 Cf. A109091, A340453, A340454, A340455, A340456.
%K A285442 sign
%O A285442 0,2
%A A285442 _Seiichi Manyama_, Apr 19 2017