A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.

1, 2, -5, -10, 9, 14, -10, 0, 14, 2, -11, -32, 0, 14, -9, 26, 2, 0, 16, -22, 14, 0, 0, 26, -17, -32, -22, -10, -34, 14, 45, 38, 0, -34, 38, -22, 2, 0, -10, 64, -20, 0, 0, 0, -23, -46, 16, 0, -46, -32, 26, -10, 25, 18, 0, 38, 50, 0, 0, -22, -80, 50, 0, 26, 2

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/12) * (eta(q^2)^5 / (eta(q) * eta(q^4)^2))^2 in powers of q.

Euler transform of period 4 sequence [ 2, -8, 2, -4, ...].

a(n) = A000727(2*n) = A187076(2*n) = A106508(4*n) = A187149(4*n).

Convolution square of A143378.

Example

G.f. = 1 + 2*x - 5*x^2 - 10*x^3 + 9*x^4 + 14*x^5 - 10*x^6 + 14*x^8 + ...
G.f. = q + 2*q^13 - 5*q^25 - 10*q^37 + 9*q^49 + 14*q^61 - 10*q^73 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A)^2))^2, n))};

Crossrefs

Cf. A000727, A106508, A143378, A187076, A187149.

Sequence in context: A323728 A140469 A001440 * A097378 A078310 A359487

Adjacent sequences: A258776 A258777 A258778 * A258780 A258781 A258782

Keyword

sign

Author

Michael Somos, Jun 09 2015

Status

approved