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A326827 Expansion of 1 / (chi(-x)^10 * chi(-x^2)^4) in powers of x where chi() is a Ramanujan theta function.

%I A326827 #19 Feb 16 2025 08:33:58
%S A326827 1,10,59,270,1045,3582,11194,32488,88716,230150,571363,1365148,
%T A326827 3153522,7069242,15425719,32849906,68421073,139645914,279740407,
%U A326827 550790788,1067244261,2037348726,3835457084,7126887974,13081454919,23735283778,42598577587,75668099822
%N A326827 Expansion of 1 / (chi(-x)^10 * chi(-x^2)^4) in powers of x where chi() is a Ramanujan theta function.
%C A326827 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A326827 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A326827 Expansion of q^(-3/4) * (eta(q^2)^3 * eta(q^4)^2 / eta(q)^5)^2 in powers of q.
%F A326827 Euler transform of period 4 sequence [10, 4, 10, 0, ...].
%F A326827 G.f.: Product_{n>=0} (1 - x^(2*n + 1))^-10 * (1 - x^(4*n + 2))^-4.
%F A326827 A093160(2*n + 1) = A123655(4*n + 3) = 4*a(n).
%F A326827 A232772(2*n + 1) = A215348(4*n + 3) = A215349(4*n + 3) = 8*a(n).
%F A326827 A007096(4*n + 3) = A212318(4*n + 3) = 16*a(n). A189925(4*n + 3) = A232358(4*n + 3) = -16*a(n).
%F A326827 a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - _Vaclav Kotesovec_, Oct 31 2019
%e A326827 G.f. = 1 + 10*x + 59*x^2 + 270*x^3 + 1045*x^4 + 3582*x^5 + 11194*x^6 + ...
%e A326827 G.f. = q^3 + 10*q^7 + 59*q^11 + 270*q^15 + 1045*q^19 + 3582*q^23 + 11194*q^27 + ...
%t A326827 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 QPochhammer[ x^4]^2 / (QPochhammer[ x]^5))^2, {x, 0, n}];
%t A326827 a[ n_] := SeriesCoefficient[ x^(-3/4) (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]^2 / 4)^2, {x, 0, n}];
%t A326827 nmax = 20; CoefficientList[Series[Product[(1 + x^k)^10/(1 - x^(4*k - 2))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 31 2019 *)
%o A326827 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 * eta(x^4 + A)^2 / eta(x + A)^5)^2, n))};
%Y A326827 Cf. A007096, A093160, A123655, A189925, A212318, A215348, A215349, A232358, A232772.
%K A326827 nonn
%O A326827 0,2
%A A326827 _Michael Somos_, Oct 20 2019