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A138521 Expansion of chi(-q)^5 / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.

%I A138521 #13 Feb 16 2025 08:33:08
%S A138521 1,-5,10,-15,30,-55,80,-120,190,-285,410,-585,840,-1190,1640,-2240,
%T A138521 3070,-4170,5570,-7400,9830,-12960,16920,-21990,28520,-36805,47180,
%U A138521 -60225,76720,-97350,122880,-154610,194110,-242880,302740,-376295,466710,-577270,711800
%N A138521 Expansion of chi(-q)^5 / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.
%C A138521 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138521 G. C. Greubel, <a href="/A138521/b138521.txt">Table of n, a(n) for n = 0..1000</a>
%H A138521 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138521 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138521 Expansion of (eta(q) / eta(q^2))^5 * eta(q^10) / eta(q^5) in powers of q.
%F A138521 Euler transform of period 10 sequence [ -5, 0, -5, 0, -4, 0, -5, 0, -5, 0, ...].
%F A138521 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * ((u+1)^2 + v) - (v + 4 * u).
%F A138521 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * v * (u - 1) * (u + 4) * (v - 1) * (v + 4).
%F A138521 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A095813.
%F A138521 G.f.: Product_{k>0} (1 + x^(5*k)) / (1 + x^k)^5.
%F A138521 a(n) = -5 * A138519(n) unless n=0. Convolution inverse of A132985.
%F A138521 a(n) = (-1)^n * A225701(n). - _Michael Somos_, Sep 15 2015
%e A138521 G.f. = 1 - 5*q + 10*q^2 - 15*q^3 + 30*q^4 - 55*q^5 + 80*q^6 - 120*q^7 + 190*q^8 + ...
%t A138521 a[ n_] := SeriesCoefficient[ QPochhammer[ -q^5, q^5] / QPochhammer[ -q, q]^5, {q, 0, n}]; (* _Michael Somos_, Sep 15 2015 *)
%o A138521 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) / eta(x^2 + A) )^5 * eta(x^10 + A) / eta(x^5 + A), n))};
%Y A138521 Cf. A095813, A132985, A138519, A225701.
%K A138521 sign
%O A138521 0,2
%A A138521 _Michael Somos_, Mar 23 2008