A132225 Expansion of (phi(q) - phi(q^5)) / (phi(q) + phi(q^5)) in powers of q where phi() is a Ramanujan theta function.

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 26.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

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.

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

Cf. A007325, A259392, A259393.

Sequence in context: A266348 A179647 A029330 * A263923 A331248 A361913

Adjacent sequences: A132222 A132223 A132224 * A132226 A132227 A132228

Keyword

sign

Author

Michael Somos, Aug 14 2007

Status

approved