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
Keywords
Examples
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 + ...
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
Programs
-
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}]; -
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))};
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) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017
Comments