A257920 Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 2, 0, 1, 4, 0, 0, 2, 0, 3, 2, 0, 2, 2, 0, 0, 2, 0, 3, 4, 0, 0, 2, 0, 0, 4, 0, 2, 2, 0, 1, 2, 0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 2, 0, 3, 4, 0, 0, 4, 0, 0, 2, 0, 4, 2, 0, 0, 2, 0, 0, 2, 0, 1, 4, 0, 2, 6, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 4, 0, 4, 2, 0, 3, 2, 0

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 q^(-3/8) * eta(q^2)^5 * eta(q^6)^2 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.

Euler transform of period 12 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].

a(n) = A129402(4*n + 1) = A134177(4*n + 1) = A000377(8*n + 3) = A192013(8*n + 3).

a(3*n + 2) = 0. a(3*n + 1) = 2 * A128591(n).

Example

G.f. = 1 + 2*x + x^3 + 4*x^4 + 2*x^7 + 3*x^9 + 2*x^10 + 2*x^12 + 2*x^13 + ...
G.f. = q^3 + 2*q^11 + q^27 + 4*q^35 + 2*q^59 + 3*q^75 + 2*q^83 + 2*q^99 + ...

Mathematica

a[ n_] := Seriescoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {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^6 + A)^2 / (eta(x + A)^2 * eta(x^3 + A) * eta(x^4 + A)^2), n))};

Crossrefs

Cf. A000377, A128591, A129402, A134177, A192013.

Sequence in context: A174996 A286815 A256276 * A258210 A258228 A271584

Adjacent sequences: A257917 A257918 A257919 * A257921 A257922 A257923

Keyword

nonn

Author

Michael Somos, Jul 12 2015

Status

approved