A256626 Expansion of psi(x) / psi(x^3) in powers of x where psi() is a Ramanujan theta function.

1, 1, 0, 0, -1, 0, 1, 1, 0, -2, -1, 0, 2, 2, 0, -2, -3, 0, 3, 3, 0, -4, -4, 0, 5, 6, 0, -6, -7, 0, 7, 8, 0, -10, -10, 0, 13, 13, 0, -14, -16, 0, 17, 18, 0, -22, -22, 0, 26, 28, 0, -30, -33, 0, 36, 38, 0, -44, -45, 0, 52, 55, 0, -60, -65, 0, 70, 74, 0, -84

Offset

0,10

Comments

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

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 f(-x^2, -x^4) / f(-x, -x^5) in powers of x where f() is Ramanujan's two-variable theta function.

Expansion of q^(1/4) * eta(q^2)^2 * eta(q^3) / (eta(q) * eta(q^6)^2) in powers of

Euler transform of period 6 sequence [ 1, -1, 0, -1, 1, 0, ...].

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4*v^2 - (v^2 - 1) * (u^4 - v^2).

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u*v - 1)^3 - (v^4 - 1).

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2 * (v^2 + w^2) - v*w * (3 + v^2).

Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = 3*u2 + u1*u3*u6 - u2^2*u6 - u1*u2*u3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A098151.

G.f.: Product_{k>0} (1 + x^k + x^(2*k))^-1 * (1 - x^k + x^(2*k))^-2.

Convolution inverse is A101195. Convolution square is A058487. Convolution 4th power is A128633.

a(n) = (-1)^n * A135211(n). a(3*n + 2) = 0.

Example

G.f. = 1 + x - x^4 + x^6 + x^7 - 2*x^9 - x^10 + 2*x^12 + 2*x^13 - 2*x^15 + ...
G.f. = 1/q + q^3 - q^15 + q^23 + q^27 - 2*q^35 - q^39 + 2*q^47 + 2*q^51 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q^(1/4) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2), n))};

Crossrefs

Cf. A058487, A098151, A101195, A128633, A135211.

Sequence in context: A271224 A157242 A281423 * A135211 A029294 A065434

Adjacent sequences: A256623 A256624 A256625 * A256627 A256628 A256629

Keyword

sign

Author

Michael Somos, Apr 05 2015

Status

approved