OFFSET
1,5
COMMENTS
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 26.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of R(q) * R(q^4) in powers of q where R(q) is the Rogers-Ramanujan continued fraction, g.f. A007325
Euler transform of period 20 sequence [ -1, 1, 1, -2, 0, -1, 1, 2, -1, 0, -1, 2, 1, -1, 0, -2, 1, 1, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 - u*v^3) * (u^3 -v) + 3 * u * v * (1 - u^2) * (1 - v^2) - 3 * u * v * (1 - u * v) * (u - v).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v, w) = (1 + u * v^3) * (u^3 + v) - (1 + u^2 * v^2) * (u^2 + v^2) - 3 * u * v * (1 + u^2) * (1 + v^2) + 5 * u * v * (1 + u * v) * (u + v) - 10 * u^2 * v^2.
Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = 13/2 + (5/2)*sqrt(5) - (1/2)*sqrt(290 + 130*sqrt(5)). - Simon Plouffe, Mar 04 2021
EXAMPLE
G.f. = q - q^2 + q^3 - 2*q^5 + 2*q^6 - 2*q^7 + q^8 + 2*q^9 - 3*q^10 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1 - 2 / (1 + EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^5]), {q, 0, n}]; (* Michael Somos, May 15 2015 *)
a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^{1, -1, -1, 2, 0, 1, -1, -2, 1, 0, 1, -2, -1, 1, 0, 2, -1, -1, 1, 0}[[Mod[k, 20, 1]]], {k, n}], {q, 0, n}]; (* Michael Somos, May 15 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod(k = 1, n, (1 - x^k + x * O(x^n))^[ 0, 1, -1, -1, 2, 0, 1, -1, -2, 1, 0, 1, -2, -1, 1, 0, 2, -1, -1, 1][k%20+1]), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 14 2007
STATUS
approved