OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of chi(x^2) / (chi(-x^12) * phi(-x)) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-5/12) * eta(q^4)^2 * eta(q^24) / (eta(q)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 2, 2, 2, 0, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 0, 2, 2, 2, 1, ...].
G.f. is a period 1 Fourier series that satisfies f(-1 / (864 t)) = 288^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A279479.
a(n) = A131945(2*n + 1).
a(n) ~ sqrt(5) * exp(sqrt(10*n)*Pi/3) / (24*n). - Vaclav Kotesovec, Nov 29 2019
EXAMPLE
G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + ...
G.f. = q^5 + 2*q^17 + 5*q^29 + 10*q^41 + 18*q^53 + 32*q^65 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^4] QPochhammer[ -x^12, x^12] / QPochhammer[ x]^2 , {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^4] QPochhammer[ -x^12, x^12] / EllipticTheta[ 4, 0, x] , {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^24 + A) / (eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q^4)^2*eta(q^24)/(eta(q)^2*eta(q^8)*eta(q^12))) \\ Altug Alkan, Jul 30 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Dec 12 2016
STATUS
approved