A328789 - OEIS

A328789

Expansion of (chi(x^3) / chi(-x^2))^2 in powers of x where chi() is a Ramanujan theta function.

1, 0, 2, 2, 3, 4, 7, 6, 11, 14, 17, 22, 32, 34, 49, 60, 72, 90, 117, 132, 171, 206, 245, 298, 369, 422, 522, 620, 728, 868, 1043, 1198, 1439, 1688, 1962, 2304, 2717, 3114, 3668, 4258, 4909, 5698, 6627, 7566, 8788, 10112, 11574, 13310, 15317, 17410, 20010

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OFFSET

0,3

COMMENTS

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

Convolution square of A097242.

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328795.

LINKS

Table of n, a(n) for n=0..50.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(1/12) * (eta(q^4) * eta(q^6)^2)^2 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

Euler transform of period 12 sequence [0, 2, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, ...].

G.f.: Product_{k>=1} (1 + x^(6*k - 3))^2 / (1 - x^(4*k - 2))^2.

a(n) = A112206(2*n).

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

EXAMPLE

G.f. = 1 + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 6*x^7 + 11*x^8 + ...

G.f. = q^-1 + 2*q^23 + 2*q^35 + 3*q^47 + 4*q^59 + 7*q^71 + 6*q^83 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2, x^2] QPochhammer[ -x^3, x^6])^2, {x, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^6 + A)^2)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

CROSSREFS

Cf. A097242, A112206, A328795.

Sequence in context: A088633 A213042 A114952 * A086969 A014692 A058670

Adjacent sequences: A328786 A328787 A328788 * A328790 A328791 A328792

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 27 2019

STATUS

approved