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%I #13 Mar 12 2021 22:24:48
%S 1,2,4,8,14,22,35,54,82,122,176,254,362,504,697,960,1307,1762,2360,
%T 3142,4158,5462,7133,9280,12013,15462,19818,25314,32198,40782,51476,
%U 64768,81226,101522,126502,157216,194846,240784,296802,365006,447794,548042,669254
%N Expansion of phi(-x^5) * f(x^2, x^8) / psi(-x)^2 in powers of x where phi, psi, f(,) are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A259392/b259392.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 Euler transform of period 20 sequence [ 2, 1, 2, 1, 0, 0, 2, 3, 2, -2, 2, 3, 2, 0, 0, 1, 2, 1, 2, 0, ...].
%F a(n) = A132225(5*n + 1).
%F a(n) ~ exp(2*Pi*sqrt(n/5)) / (5^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 04 2018
%e G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 14*x^4 + 22*x^5 + 35*x^6 + 54*x^7 + ...
%e G.f. = q + 2*q^6 + 4*q^11 + 8*q^16 + 14*q^21 + 22*q^26 + 35*q^31 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^10] QPochhammer[ -x^8, x^10] (QPochhammer[ x^2, x^4] QPochhammer[ x^5] / QPochhammer[ x])^2, {x, 0, n}];
%t QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y]; A:= f[-x^5, -x^5]*f[x^2, x^8]/f[-x, - x^3]^2; a:= CoefficientList[Series[A, {x,0,60}], x]; Table[a[[n]], {n,1,50}] (* _G. C. Greubel_, Jul 04 2018 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, -2, -1, -2, -1, 0, 0, -2, -3, -2, 2, -2, -3, -2, 0, 0, -1, -2, -1, -2][k%20 + 1]), n))};
%Y Cf. A132225.
%K nonn
%O 0,2
%A _Michael Somos_, Jun 25 2015
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