A263433 Expansion of f(x, x) * f(x^2, x^4)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 2, 4, 5, 6, 6, 4, 7, 4, 6, 8, 4, 10, 8, 12, 8, 6, 14, 8, 11, 6, 8, 8, 8, 14, 6, 12, 15, 14, 14, 8, 12, 14, 12, 16, 8, 10, 14, 16, 16, 12, 12, 12, 16, 10, 10, 8, 19, 20, 20, 8, 12, 24, 14, 24, 12, 16, 14, 16, 21, 10, 14, 28, 16, 12, 14, 12, 16, 16, 30, 12

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 f(-x^2)^2 * phi(-x^6)^2 / phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 15552^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263444.

a(n) = A261426(2*n) = A045832(6*n). 3 * a(n) = A005889(6*n).

Example

G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 6*x^6 + 4*x^7 + 7*x^8 + ...
G.f. = q + 2*q^7 + 2*q^13 + 4*q^19 + 5*q^25 + 6*q^31 + 6*q^37 + 4*q^43 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 4, 0, x], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A)^4 / (eta(x + A)^2 * eta(x^12 + A)^2), n))};

Crossrefs

Cf. A005889, A045832, A261426, A263444.

Sequence in context: A353847 A181537 A130498 * A147806 A338228 A351782

Adjacent sequences: A263430 A263431 A263432 * A263434 A263435 A263436

Keyword

nonn

Author

Michael Somos, Oct 18 2015

Status

approved