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%I #16 Mar 12 2021 22:24:48
%S 1,1,2,3,5,7,11,15,20,28,38,50,67,87,113,146,186,236,299,375,468,583,
%T 721,888,1093,1336,1628,1980,2397,2894,3487,4186,5013,5991,7139,8488,
%U 10073,11924,14086,16613,19551,22965,26934,31527,36844,42994,50085,58258
%N Expansion of f(-x^8)^2 / f(-x) 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 Seiichi Manyama, <a href="/A260164/b260164.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 q^(-5/8) * eta(q^8)^2 / eta(q) in powers of q.
%F Euler transform of period 8 sequence [ 1, 1, 1, 1, 1, 1, 1, -1, ...].
%F 2 * a(n) = A132965(2*n + 1).
%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 20*x^8 + ...
%e G.f. = q^5 + q^13 + 2*q^21 + 3*q^29 + 5*q^37 + 7*q^45 + 11*q^53 + 15*q^61 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 / QPochhammer[ x], {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 / eta(x + A), n))};
%Y Cf. A010054, A132965.
%K nonn
%O 0,3
%A _Michael Somos_, Nov 09 2015
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