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A143895 Expansion of (chi(q)^4 / chi(-q))^2 in powers of q where chi() is a Ramanujan theta function.

%I A143895 #16 Feb 16 2025 08:33:08
%S A143895 1,10,47,150,403,1002,2316,5004,10309,20456,39240,73102,132779,235868,
%T A143895 410785,702630,1182342,1960418,3206675,5179670,8270086,13062994,
%U A143895 20427293,31644200,48589970,73994118,111802523,167685238,249745021,369499928
%N A143895 Expansion of (chi(q)^4 / chi(-q))^2 in powers of q where chi() is a Ramanujan theta function.
%C A143895 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A143895 G. C. Greubel, <a href="/A143895/b143895.txt">Table of n, a(n) for n = 0..1000</a>
%H A143895 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143895 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143895 Expansion of q^(1/4) * (eta(q^2)^9 / (eta(q)^5 * eta(q^4)^4))^2 in powers of q.
%F A143895 Euler transform of period 4 sequence [ 10, -8, 10, 0, ...].
%F A143895 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143894.
%F A143895 G.f.: (Product_{k>0} (1 + x^k)^5 / (1 + x^(2*k))^4)^2.
%F A143895 a(n) ~ exp(sqrt(2*n)*Pi) / (2^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%e A143895 G.f. = 1 + 10*x + 47*x^2 + 150*x^3 + 403*x^4 + 1002*x^5 + 2316*x^6 + 5004*x^7 + ...
%e A143895 G.f. = 1/q + 10*q^3 + 47*q^7 + 150*q^11 + 403*q^15 + 1002*q^19 + 2316*q^23 + ...
%t A143895 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^9 / (QPochhammer[ x]^5 QPochhammer[ x^4]^4))^2, {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%t A143895 nmax = 40; CoefficientList[Series[Product[((1 + x^k)^5 / (1 + x^(2*k))^4)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o A143895 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^9 / (eta(x + A)^5 * eta(x^4 + A)^4))^2, n))};
%Y A143895 Cf. A143894.
%K A143895 nonn
%O A143895 0,2
%A A143895 _Michael Somos_, Sep 04 2008