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%I #15 Mar 12 2021 22:24:45
%S 1,8,26,48,79,168,326,496,755,1296,2106,3072,4460,6840,10284,14448,
%T 20165,29184,41640,56880,77352,107472,147902,197616,263019,354888,
%U 475516,624048,816065,1076736,1413142,1826416,2353446,3050400,3936754,5022720
%N Expansion of (chi(q)^5 * chi(-q))^2 in powers of q 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="/A143894/b143894.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^(1/2) * (eta(q^2)^9 / (eta(q)^4 * eta(q^4)^5))^2 in powers of q.
%F Euler transform of period 4 sequence [ 8, -10, 8, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143895.
%F G.f.: (Product_{k>0} (1 + x^k)^4 / (1 + x^(2*k))^5)^2.
%F a(2*n) = A052241(n). a(2*n + 1) = 8 * A022571(n).
%F a(n) ~ exp(sqrt(n)*Pi) / (sqrt(2) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%e G.f. = 1 + 8*x + 26*x^2 + 48*x^3 + 79*x^4 + 168*x^5 + 326*x^6 + 496*x^7 + ...
%e G.f. = 1/q + 8*q + 26*q^3 + 48*q^5 + 79*q^7 + 168*q^9 + 326*q^11 + 496*q^13 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^9 / (QPochhammer[ x]^4 QPochhammer[ x^4]^5))^2, {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%t nmax = 40; CoefficientList[Series[Product[((1 + x^k)^4 / (1 + x^(2*k))^5)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 / (eta(x + A)^4 * eta(x^4 + A)^5))^2, n))};
%Y Cf. A022571, A052241, A143895.
%K nonn
%O 0,2
%A _Michael Somos_, Sep 04 2008
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