OFFSET
1,2
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q * (psi(q^3) / psi(q))^4 in powers of q where psi() is a Ramanujan theta function.
Expansion of ((eta(q^6) / eta(q^2))^2 * (eta(q) / eta(q^3)))^4 in powers of q.
Euler transform of period 6 sequence [ -4, 4, 0, 4, -4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1-u) * (1-9*u) - (u-v)^2.
G.f.: x * (Product_{k>0} (1 - x^k + x^(2*k))^2 * (1 + x^k + x^(2*k)) )^4.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (81*u^2*v^2 + 9*u*v - 12*u + 30*u^2 - 108*u^2*v + 1) * v - u^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/9) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128637.
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/3 + (2/9)*sqrt(3) - (2/9)*sqrt(6)*3^(1/4). - Simon Plouffe, Mar 02 2021
EXAMPLE
G.f. = q - 4*q^2 + 10*q^3 - 20*q^4 + 39*q^5 - 76*q^6 + 140*q^7 - 244*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(3/2)] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ((eta(x^6 + A) / eta(x^2 + A))^2 * eta(x + A) / eta(x^3 + A))^4, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 16 2007
STATUS
approved