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A132225
Expansion of (phi(q) - phi(q^5)) / (phi(q) + phi(q^5)) in powers of q where phi() is a Ramanujan theta function.
3
1, -1, 1, 0, -2, 2, -2, 1, 2, -3, 4, -4, 1, 2, -5, 8, -7, 3, 2, -10, 14, -12, 6, 6, -17, 22, -20, 8, 10, -26, 35, -31, 12, 14, -39, 54, -47, 20, 20, -61, 82, -72, 31, 32, -93, 122, -107, 44, 50, -133, 176, -154, 61, 68, -189, 254, -220, 90, 94, -272, 362
OFFSET
1,5
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 26.
LINKS
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.
a(5*n) = -A259393(n) unless n=0. a(5*n + 1) = A259392(n). - Michael Somos, Jun 25 2015
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