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

0,2

Comments

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

Links

Table of n, a(n) for n=0..27.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Example

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 + ...

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 *)

Prog

(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))};

Crossrefs

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

Sequence in context: A226796 A061001 A055586 * A054489 A219580 A267021

Adjacent sequences: A326824 A326825 A326826 * A326828 A326829 A326830

Keyword

nonn

Author

Michael Somos, Oct 20 2019

Status

approved