A263993 Expansion of f(-x, x^2) / f(-x, -x^3)^3 in powers of x where f(, ) is Ramanujan's general theta function.

1, 2, 4, 10, 20, 36, 64, 112, 189, 308, 492, 778, 1210, 1844, 2776, 4144, 6114, 8914, 12884, 18484, 26302, 37124, 52040, 72512, 100415, 138196, 189160, 257648, 349184, 470932, 632312, 845472, 1125853, 1493222, 1973060, 2597892, 3408754, 4457600, 5810544

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^3) / (phi(-x) * f(-x^4)^2) in powers of x where phi(), f() are Ramanujan theta functions.

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

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

a(n) = A133637(3*n - 1).

Example

G.f. = 1 + 2*x + 4*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 64*x^6 + 112*x^7 + ...
G.f. = 1/q + 2*q^2 + 4*q^5 + 10*q^8 + 20*q^11 + 36*q^14 + 64*q^17 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^5 / (eta(x + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^12 + A)^2), n))};

Crossrefs

Cf. A133637.

Sequence in context: A127392 A236001 A258092 * A189585 A239346 A004647

Adjacent sequences: A263990 A263991 A263992 * A263994 A263995 A263996

Keyword

nonn

Author

Michael Somos, Oct 31 2015

Status

approved