OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (phi(q^5) / phi(q))^2 * (chi^5(q) / chi(q^5)) in powers of q where phi(), chi() are Ramanujan theta functions.
Expansion of eta(q^10)^8 / (eta(q) * eta(q^4) * eta(q^5)^3 * eta(q^20)^3) in powers of q.
Euler transform of period 20 sequence [ 1, 1, 1, 2, 4, 1, 1, 2, 1, -4, 1, 2, 1, 1, 4, 2, 1, 1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A225849.
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
EXAMPLE
G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 11*x^5 + 16*x^6 + 24*x^7 + 38*x^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 + (EllipticTheta[ 2, Pi/4, q^(5/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)])^2, {q, 0, n}]; (* Michael Somos, Oct 26 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^5] / EllipticTheta[ 3, 0, q])^2 QPochhammer[ q^5, -q^5] / QPochhammer[ q, -q]^5, {q, 0, n}]; (* Michael Somos, Oct 26 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A)^8 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A)^3 * eta(x^20 + A)^3), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 05 2013
STATUS
approved