OFFSET
0,2
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(5/3) * (eta(q) * eta(q^19) / (eta(q^2) * eta(q^38)))^2 in powers of q.
Euler transform of period 38 sequence [ -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -4, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (342 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134004.
G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(19*k)))^-2.
a(n) ~ (-1)^n * 5^(1/4) * exp(2*Pi*sqrt(5*n/57)) / (2 * 57^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
EXAMPLE
G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^4 - 4*x^5 + 5*x^6 - 6*x^7 + 9*x^8 - 12*x^9 + ...
G.f. = q^-5 - 2*q^-2 + q - 2*q^4 + 4*q^7 - 4*q^10 + 5*q^13 - 6*q^16 + 9*q^19 - ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x, x^2] QPochhammer[ x^19, x^38])^2, {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^19 + A) / (eta(x^2 + A) * eta(x^38 + A)))^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 01 2007
STATUS
approved