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%I #15 Mar 12 2021 22:24:47
%S 1,1,6,15,26,57,102,172,276,453,743,1128,1698,2539,3780,5531,7882,
%T 11238,15918,22259,30861,42438,58110,78909,106392,142872,190698,
%U 253179,334266,439581,575956,750613,974316,1260336,1624702,2086806,2670162,3406695,4333590
%N Expansion of chi(x^5)^6 + x * chi(x)^6 in powers of x 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 G. C. Greubel, <a href="/A240948/b240948.txt">Table of n, a(n) for n = 0..1000</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 (chi(x) * chi(x^5))^6 - 4 * x * chi(x) * chi(x^5) in powers of x where chi() is a Ramanujan theta function.
%F Expansion of (eta(q^2)^2 / (eta(q) * eta(q^4)))^6 + (eta(q^10)^2 / (eta(q^5) * eta(q^20)))^6 in powers of q.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (320 t)) = f(t) where q = exp(2 Pi i t).
%F a(n) ~ exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Nov 29 2019
%e G.f. = 1 + x + 6*x^2 + 15*x^3 + 26*x^4 + 57*x^5 + 102*x^6 + 172*x^7 + ...
%e G.f. = q^-5 + q^-1 + 6*q^3 + 15*q^7 + 26*q^11 + 57*q^15 + 102*q^19 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^5, -x^5]^-6 + x QPochhammer[ x, -x]^-6, {x, 0, n}];
%t nmax = 40; CoefficientList[Series[x*Product[((1 + x^k)/(1 + x^(2*k)))^6, {k, 1, nmax}] + Product[((1 + x^(5*k))/(1 + x^(10*k)))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 29 2019 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( x * (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^6 + (eta(x^10 + A)^2 / (eta(x^5 + A) * eta(x^20 + A)))^6, n))};
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = eta(x^2 + A)^2 * eta(x^10 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)); polcoeff( A^5 - 4 * x * A, n))};
%K nonn
%O 0,3
%A _Michael Somos_, Aug 04 2014
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