%I #14 Sep 08 2022 08:46:08
%S 1,2,5,10,20,26,45,60,85,100,156,172,240,270,365,376,517,520,705,740,
%T 932,942,1260,1200,1510,1580,1928,1880,2420,2300,2861,2864,3410,3310,
%U 4265,3876,4780,4740,5625,5300,6672,6082,7460,7270,8400,8026,10092,9100
%N Expansion of f(-x^5)^10 / f(-x)^2 in powers of x where f() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A243938/b243938.txt">Table of n, a(n) for n = 0..2500</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 q^(-2) * eta(q^5)^10 / eta(q)^2 in powers of q.
%F Euler transform of period 5 sequence [2, 2, 2, 2, -8, ...].
%F Given g.f. A(x), then B(q) = q^2 * A(q) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (v^3 + u^2*w + 16*u*w^2)^2 - 4*u*w * (u + 2*v) * (v + 8*w) * (v^2 + 2*u*w).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 1/5 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A243939.
%F Convolution square of A053723.
%e G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 26*x^5 + 45*x^6 + 60*x^7 + ...
%e G.f. = q^2 + 2*q^3 + 5*q^4 + 10*q^5 + 20*q^6 + 26*q^7 + 45*q^8 + 60*q^9 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^5]^10 / QPochhammer[ x]^2, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^5 + A)^10 / eta(x + A)^2, n))};
%o (Magma) Basis( ModularForms( Gamma0(5), 4), 49) [3];
%o (Sage) A = ModularForms( Gamma0(5), 4, prec=49) . basis(); (-A[0] + A[2]) / 13;
%Y Cf. A053723, A243939.
%K nonn
%O 0,2
%A _Michael Somos_, Jun 15 2014