A262368 Expansion of f(x^2, x^2) * f(x, x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 5, 6, 7, 6, 4, 8, 8, 14, 11, 8, 8, 6, 15, 14, 12, 12, 8, 14, 16, 12, 16, 10, 19, 20, 12, 14, 12, 14, 21, 14, 16, 14, 16, 30, 12, 20, 16, 20, 20, 16, 24, 12, 25, 26, 16, 16, 16, 28, 16, 14, 25, 18, 24, 30, 20, 16, 28, 38, 32, 14, 16, 22, 20, 28, 28, 16

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

Expansion of q^(-1/12) * eta(q^3)^4 * eta(q^4)^5 / (eta(q)^2 * eta(q^6)^2 * eta(q^8)^2) in powers of q.

Euler transform of period 24 sequence [ 2, 2, -2, -3, 2, 0, 2, -1, -2, 2, 2, -5, 2, 2, -2, -1, 2, 0, 2, -3, -2, 2, 2, -3, ...].

a(n) = A261426(4*n).

Example

G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 7*x^4 + 6*x^5 + 4*x^6 + 8*x^7 + ...
G.f. = q + 2*q^13 + 5*q^25 + 6*q^37 + 7*q^49 + 6*q^61 + 4*q^73 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^4 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^6 + A)^2 * eta(x^8 + A)^2), n))};

Crossrefs

Cf. A261426.

Sequence in context: A225366 A146099 A004097 * A198547 A120678 A078384

Adjacent sequences: A262365 A262366 A262367 * A262369 A262370 A262371

Keyword

nonn

Author

Michael Somos, Oct 02 2015

Status

approved