OFFSET
0,3
COMMENTS
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (first 2501 terms from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (phi(x) / psi(x^2)^2) * (phi(x) - phi(x^2)) / (2*x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1) * phi(q^2) * (phi(q^2) - phi(q^4)) / (phi(q^4) * (phi(q) - phi(q^4))) in powers of q^2 where phi() is a Ramanujan theta function.
Euler transform of period 8 sequence [1, -5, 3, 2, 3, -5, 1, 0, ...].
a(2*n + 1) = A226559(n).
EXAMPLE
G.f. = 1 + x - 4*x^2 - x^3 + 11*x^4 - x^5 - 20*x^6 + 2*x^7 + 34*x^8 + ...
G.f. = q + q^3 - 4*q^5 - q^7 + 11*q^9 - q^11 - 20*q^13 + 2*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2 q^(-1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q]^2 (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]), {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^6) * sum( k=1, sqrtint(n+1), x^k^2 - x^(2*k^2), x * A) / x, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 22 2013
STATUS
approved