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 (chi(x) * chi(x^5))^6 - 4 * x * chi(x) * chi(x^5) in powers of x where chi() is a Ramanujan theta function.
Expansion of (eta(q^2)^2 / (eta(q) * eta(q^4)))^6 + (eta(q^10)^2 / (eta(q^5) * eta(q^20)))^6 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (320 t)) = f(t) where q = exp(2 Pi i t).
a(n) ~ exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Nov 29 2019
EXAMPLE
G.f. = 1 + x + 6*x^2 + 15*x^3 + 26*x^4 + 57*x^5 + 102*x^6 + 172*x^7 + ...
G.f. = q^-5 + q^-1 + 6*q^3 + 15*q^7 + 26*q^11 + 57*q^15 + 102*q^19 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^5, -x^5]^-6 + x QPochhammer[ x, -x]^-6, {x, 0, n}];
nmax = 40; CoefficientList[Series[x*Product[((1 + x^k)/(1 + x^(2*k)))^6, {k, 1, nmax}] + Product[((1 + x^(5*k))/(1 + x^(10*k)))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2019 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( x * (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^6 + (eta(x^10 + A)^2 / (eta(x^5 + A) * eta(x^20 + A)))^6, n))};
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = eta(x^2 + A)^2 * eta(x^10 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)); polcoeff( A^5 - 4 * x * A, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 04 2014
STATUS
approved