A258278 Expansion of f(-x, -x^5)^2 in powers of x where f(,) is Ramanujan's general theta function.

1, -2, 1, 0, 0, -2, 2, 0, 2, -2, 1, 0, 0, -2, 0, 0, 3, -2, 0, 0, 0, -4, 2, 0, 2, 0, 2, 0, 0, -2, 0, 0, 1, -2, 2, 0, 0, -2, 2, 0, 2, -4, 1, 0, 0, -2, 0, 0, 2, -2, 0, 0, 0, 0, 2, 0, 4, -2, 0, 0, 0, -4, 0, 0, 2, -2, 3, 0, 0, 0, 2, 0, 2, -4, 0, 0, 0, -2, 0, 0, 1

Offset

0,2

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 (chi(-x) * psi(x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.

Expansion of q^(-2/3) * (eta(q) * eta(q^6)^2 / (eta(q^2) * eta(q^3)))^2 in powers of q.

Euler transform of period 6 sequence [ -2, 0, 0, 0, -2, -2, ...].

a(n) = (-1)^n * A122856(n). Convolution square of A089802.

a(2*n) = A122865(n). a(4*n +1) = -2 * A121444(n). a(4*n + 2) = A122856(n). a(4*n + 3) = a(8*n + 4) = 0. a(8*n) = A002175(n). a(8*n + 2) = A122856(n).

2 * a(n) = -A258210(3*n + 2). - Michael Somos, May 01 2016

Example

G.f. = 1 - 2*x + x^2 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + x^10 - 2*x^13 + ...
G.f. = q^2 - 2*q^5 + q^8 - 2*q^17 + 2*q^20 + 2*q^26 - 2*q^29 + q^32 + ...

Mathematica

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

Prog

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

Crossrefs

Cf. A002175, A089802, A121444, A122856, A122865, A258210.

Sequence in context: A123484 A008626 A058626 * A122856 A328797 A328795

Adjacent sequences: A258275 A258276 A258277 * A258279 A258280 A258281

Keyword

sign

Author

Michael Somos, May 25 2015

Status

approved