A138527 Expansion of phi(-q) / phi(-q^5) in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, 0, 2, 2, -4, 0, 0, 2, 4, -8, 0, 0, 4, 8, -14, 0, 0, 8, 14, -24, 0, 0, 12, 22, -40, 0, 0, 20, 36, -64, 0, 0, 32, 56, -98, 0, 0, 48, 84, -148, 0, 0, 72, 126, -220, 0, 0, 106, 184, -320, 0, 0, 152, 264, -460, 0, 0, 216, 376, -652, 0, 0, 306, 528, -912, 0, 0, 424, 732, -1264, 0, 0, 584, 1008

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Denoted by t in Andrews and Berndt 2005. - Michael Somos, Apr 25 2016

References

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 337.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q) / eta(q^5))^2 * eta(q^10) / eta(q^2) in powers of q.

Euler transform of period 10 sequence [ -2, -1, -2, -1, 0, -1, -2, -1, -2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - u^2)^2 - u^2 * (1 - v^2) * (5 - v^2).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (1 - u^2) * (5 - v^2).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u + v)^2 * w^2 - u * v * (5 - v^2).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u2 * u3 - u1 * u6)^2 - u1 * u3 * (u6^2 - u2^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A116494.

G.f.: Product_{k>0} P(10, x^k) / P(5, x^k) where P(n, x) is the n-th cyclotomic polynomial.

a(5*n + 2) = a(5*n + 3) = 0.

Convolution inverse is A138526. Convolution square is A138518.

Example

G.f. = 1 - 2*q + 2*q^4 + 2*q^5 - 4*q^6 + 2*q^9 + 4*q^10 - 8*q^11 + 4*q^14 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^5], {q, 0, n}]; (* Michael Somos, Sep 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^2 * eta(x^10 + A) / eta(x^2 + A), n))};

Crossrefs

Cf. A116494, A138158, A138526.

Sequence in context: A048866 A262904 A144377 * A033718 A033737 A033747

Adjacent sequences: A138524 A138525 A138526 * A138528 A138529 A138530

Keyword

sign

Author

Michael Somos, Mar 23 2008

Status

approved