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A285932 Expansion of (Product_{k>0} (1 - x^k) / (1 - x^(5*k)))^5 in powers of x.

%I A285932 #21 Feb 16 2025 08:33:44
%S A285932 1,-5,5,10,-15,-1,-30,50,65,-95,-1,-170,220,300,-380,0,-635,820,1025,
%T A285932 -1310,0,-2045,2525,3140,-3845,2,-5780,7070,8565,-10405,-1,-15130,
%U A285932 18125,21760,-25960,0,-36820,43780,51785,-61290,0,-85170,100030,117500,-137550,0
%N A285932 Expansion of (Product_{k>0} (1 - x^k) / (1 - x^(5*k)))^5 in powers of x.
%C A285932 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A285932 Seiichi Manyama, <a href="/A285932/b285932.txt">Table of n, a(n) for n = 0..10000</a>
%H A285932 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A285932 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A285932 a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A116073(k)*a(n-k) for n > 0.
%F A285932 Expansion of q^(5/6) * eta(q)^5 / eta(q^5)^5 in powers of q. - _Michael Somos_, Apr 29 2017
%F A285932 Expansion of f(-x)^5 / f(-x^5)^5 in powers of x where f() is a Ramanujan theta function. - _Michael Somos_, Apr 29 2017
%F A285932 Euler transform of period 5 sequence [-5, -5, -5, -5, 0, ...]. - _Michael Somos_, Apr 29 2017
%e A285932 G.f. = 1 - 5*x + 5*x^2 + 10*x^3 - 15*x^4 - x^5 - 30*x^6 + 50*x^7 + 65*x^8 - 95*x^9 + ...
%e A285932 G.f. = q^-5 - 5*q + 5*q^7 + 10*q^13 - 15*q^19 - q^25 - 30*q^31 + 50*q^37 + 65*q^43 + ...
%t A285932 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^5 / QPochhammer[ x^5]^5, {x, 0, n}]; (* _Michael Somos_, Apr 29 2017 *)
%o A285932 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^5, n))}; /* _Michael Somos_, Apr 29 2017 */
%Y A285932 (Product_{k>0} (1 - x^k) / (1 - x^(m*k)))^m: A022597 (m=2), A199659 (m=3), A112143 (m=4), this sequence (m=5).
%Y A285932 Cf. A285928.
%K A285932 sign
%O A285932 0,2
%A A285932 _Seiichi Manyama_, Apr 29 2017