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

1, 10, 59, 270, 1045, 3582, 11194, 32488, 88716, 230150, 571363, 1365148, 3153522, 7069242, 15425719, 32849906, 68421073, 139645914, 279740407, 550790788, 1067244261, 2037348726, 3835457084, 7126887974, 13081454919, 23735283778, 42598577587, 75668099822

Offset: 0

Author

Michael Somos, Oct 20 2019

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 10*x + 59*x^2 + 270*x^3 + 1045*x^4 + 3582*x^5 + 11194*x^6 + ...
G.f. = q^3 + 10*q^7 + 59*q^11 + 270*q^15 + 1045*q^19 + 3582*q^23 + 11194*q^27 + ...
		

Links

• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A007096, A093160, A123655, A189925, A212318, A215348, A215349, A232358, A232772.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 QPochhammer[ x^4]^2 / (QPochhammer[ x]^5))^2, {x, 0, n}];
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}];
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 *)

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))};

Formula

Expansion of q^(-3/4) * (eta(q^2)^3 * eta(q^4)^2 / eta(q)^5)^2 in powers of q.

Euler transform of period 4 sequence [10, 4, 10, 0, ...].

G.f.: Product_{n>=0} (1 - x^(2*n + 1))^-10 * (1 - x^(4*n + 2))^-4.

A093160(2*n + 1) = A123655(4*n + 3) = 4*a(n).

A232772(2*n + 1) = A215348(4*n + 3) = A215349(4*n + 3) = 8*a(n).

A007096(4*n + 3) = A212318(4*n + 3) = 16*a(n). A189925(4*n + 3) = A232358(4*n + 3) = -16*a(n).

a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019