A260150 Expansion of f(x, x^5)^3 / (f(-x, -x^5) * f(-x^2, -x^2)^2) in powers of x where f(, ) is Ramanujan's general theta function.

1, 4, 11, 24, 48, 92, 170, 304, 526, 884, 1451, 2336, 3700, 5772, 8876, 13472, 20207, 29988, 44072, 64184, 92680, 132760, 188758, 266512, 373838, 521152, 722266, 995432, 1364684, 1861548, 2527224, 3415344, 4595497, 6157700, 8218050, 10925848, 14472520

Offset

0,2

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(x) * psi(-x^3)^3 / (f(-x)^3 * psi(x^3)) in powers of x where psi(), f() are Ramanujan theta functions.

Euler transform of period 12 sequence [ 4, 1, 0, 2, 4, 2, 4, 2, 0, 1, 4, 0, ...].

a(n) = (-1)^n * A260057(n). a(n) = A261154(3*n + 2). a(2*n + 1) = 4 * A259033(n).

a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 + 4*x + 11*x^2 + 24*x^3 + 48*x^4 + 92*x^5 + 170*x^6 + 304*x^7 + ...
G.f. = q^2 + 4*q^5 + 11*q^8 + 24*q^11 + 48*q^14 + 92*q^17 + 170*q^20 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) QPochhammer[ -x] / QPochhammer[x]^3 EllipticTheta[ 2, Pi/4, x^(3/2)]^3 / EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^4 * eta(x^12 + A)^3 / (eta(x + A)^4 * eta(x^4 + A) * eta(x^6 + A)^5), n))};

Crossrefs

Cf. A259033, A260057, A261154.

Sequence in context: A290707 A322618 A260057 * A258472 A007678 A339493

Adjacent sequences: A260147 A260148 A260149 * A260151 A260152 A260153

Keyword

nonn

Author

Michael Somos, Nov 08 2015

Status

approved