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%I #23 Mar 12 2021 22:24:47
%S 1,5,10,5,-15,-24,15,70,30,-125,-175,95,420,180,-615,-826,410,1760,
%T 705,-2415,-3100,1530,6270,2460,-8090,-10174,4840,19570,7500,-24360,
%U -30024,14130,55970,21155,-67380,-81926,37895,148410,55305,-174500,-209577,96025
%N The expansion of R(q)^-5 in powers of q where R() is the Rogers-Ramanujan continued fraction.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Number 8 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Jul 22 2014
%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(5). [Yang 2004] - _Michael Somos_, Jul 22 2014
%H Seiichi Manyama, <a href="/A229793/b229793.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/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%H Y. Yang, <a href="http://dx.doi.org/10.1112/S0024609304003510">Transformation formulas for generalized Dedekind eta functions</a>, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
%F Expansion of (1/q) * (f(-q^2, -q^3) / f(-q, -q^4))^5 in powers of q where f() is a Ramanujan theta function.
%F Euler transform of period 5 sequence [ 5, -5, -5, 5, 0, ...].
%F G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = u^2 + v - u*v^2 * (u^2 - v) + 10 * u*v * (1 + u - v + u*v).
%F G.f.: (1/x) * exp( integrate_{0..x} (1 - eta(x)^5 / eta(x^5)) dx / x ).
%F G.f.: (1/x) * (Product_{k>0} (1 - x^{5*k - 2}) * (1 - x^{5*k - 3}) / (1 - x^{5*k - 1}) * (1 - x^{5*k - 4}))^5.
%F G.f.: (1/x) * ( (Sum_{k in Z} (-1)^n * x^((5*k + 1) * k/2)) / (Sum_{k in Z} (-1)^n * x^((5*k + 3) * k/2)))^5.
%F Convolution inverse of A078905.
%F a(-1) = 1, a(n) = (5/(n+1))*Sum_{k=1..n+1} A109091(k)*a(n-k) for n > -1. - _Seiichi Manyama_, Apr 01 2017
%e G.f. = 1/q + 5 + 10*q + 5*q^2 - 15*q^3 - 24*q^4 + 15*q^5 + 70*q^6 + ...
%t a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5] / QPochhammer[ q, q^5] QPochhammer[ q^4, q^5])^5, {q, 0, n}];
%o (PARI) {a(n) = local(A); if( n<-1, 0, A = x^2 * O(x^n); A = (eta(x + A) / eta(x^5 + A))^6 / x; polcoeff( (11 + A + sqrt(125 + 22*A + A^2)) / 2, n))};
%Y Cf. A003823, A078905, A109064.
%K sign
%O -1,2
%A _Michael Somos_, Sep 29 2013
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