A093085 Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

1, -2, 0, 0, 1, 2, 0, 0, -1, -4, 0, 0, 0, 6, 0, 0, 1, -8, 0, 0, 0, 12, 0, 0, -1, -18, 0, 0, -1, 24, 0, 0, 2, -32, 0, 0, 1, 44, 0, 0, -2, -58, 0, 0, -1, 76, 0, 0, 2, -100, 0, 0, 1, 128, 0, 0, -3, -164, 0, 0, -1, 210, 0, 0, 4, -264, 0, 0, 2, 332, 0, 0, -5, -416, 0, 0, -2, 516, 0, 0, 5, -640, 0, 0, 2, 790, 0, 0, -6, -968

Offset

0,2

Comments

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

eta(q^2) * eta(q^8)^6 = eta(q)^2 * eta(q^4)^2 * eta(q^8) * eta(q^16)^2 + 2 * eta(q^2) * eta(q^4)^2 * eta(q^16)^4 is equivalent to the a(4*n), ..., a(4*n + 3) results.

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 q^(1/2) * eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2) in powers of q.

Euler transform of period 8 sequence [ -2, -1, -2, -2, -2, -1, -2, 0, ...].

Given g.f. A(x), then B(q) = A(q)^2 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187154.

G.f.: Product_{k>0} (1 - x^k)^2 / ((1 - x^(4*k - 2)) * (1 - x^(8*k))^2).

a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) = -2 * A083365(n). Convolution square is A131124. Convolution inverse is A187154.

Example

G.f. = 1 - 2*x + x^4 + 2*x^5 - x^8 - 4*x^9 + 6*x^13 + x^16 - 8*x^17 + 12*x^21 - ...
G.f. = 1/q - 2*q + q^7 + 2*q^9 - q^15 - 4*q^17 + 6*q^25 + q^31 - 8*q^33 + ...

Mathematica

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

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 2, 1, 2, 2, 2, 1, 2][1 + k%8], 1 + x * O(x^n)), n))}
(PARI) {a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = 4*A + 16*A^2 + (1 + 8*A) * sqrt(A + 4*A^2)); polcoeff( sqrt(x / A), n))}
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2), n))}

Crossrefs

Cf. A029838, A029839, A083365, A131124, A187154.

Sequence in context: A208589 A112171 A112172 * A339059 A023555 A357704

Adjacent sequences: A093082 A093083 A093084 * A093086 A093087 A093088

Keyword

sign

Author

Michael Somos, Mar 20 2004, Oct 22 2007

Status

approved