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%I #24 Mar 11 2021 18:18:26
%S 1,-1,0,-1,1,4,-4,-1,-3,3,12,-12,-2,-8,8,31,-30,-5,-20,19,72,-68,-12,
%T -44,41,154,-144,-24,-90,84,312,-289,-48,-178,164,603,-554,-92,-336,
%U 307,1122,-1024,-168,-612,557,2024,-1836,-300,-1087,983,3552,-3206,-522,-1880,1692,6088,-5472,-886,-3180
%N Expansion of q * chi(-q) / chi(-q^5)^5 in powers of q where chi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Seiichi Manyama, <a href="/A095813/b095813.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of (1 - (phi(-q) / phi(-q^5))^2) / 4 in powers of q where phi() is a Ramanujan theta function.
%F Expansion of (eta(q) * eta(q^10)^5) / (eta(q^2) * eta(q^5)^5) in powers of q.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + 2*u*v + 4*u*v^2.
%F G.f. A(x) satisfies A(x^2) = -A(x) * A(-x).
%F Euler transform of period 10 sequence [ -1, 0, -1, 0, 4, 0, -1, 0, -1, 0, ...].
%F G.f.: x * (Product_{k>=1} ((1 - x^k) * (1-x^(10*k))^5) / ((1 - x^(2*k)) * (1 - x^(5*k))^5)).
%F a(n) = A138522(n) unless n = 0. Convolution inverse is A132980.
%F Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = -13/8 - (5/8)*sqrt(5) + (5/8)*sqrt(10 + 6*sqrt(5)). - _Simon Plouffe_, Mar 01 2021
%e q - q^2 - q^4 + q^5 + 4*q^6 - 4*q^7 - q^8 - 3*q^9 + 3*q^10 + 12*q^11 + ...
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^5 / (eta(x^2 + A) * eta(x^5 + A)^5), n))}
%Y Cf. A132980, A138522.
%K sign
%O 1,6
%A _Michael Somos_, Jun 07 2004
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