A260221 Expansion of phi(x^3)^2 / f(x) in powers of x where phi(), f() are Ramanujan theta functions.

1, -1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 4, 5, 3, 7, 8, 7, 7, 9, 10, 11, 12, 14, 17, 19, 18, 24, 26, 26, 31, 36, 38, 41, 45, 50, 57, 61, 63, 75, 83, 86, 93, 106, 115, 123, 134, 146, 162, 173, 183, 206, 225, 237, 257, 283, 304, 327, 350, 380, 416, 443, 471, 516, 557

Offset

0,3

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

Expansion of q^(1/24) * eta(q) * eta(q^4) * eta(q^6)^10 / (eta(q^2)^3 * eta(q^3)^4 * eta(q^12)^4) in powers of q.

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

a(n) = A259538(3*n).

Example

G.f. = 1 - x + 2*x^2 + x^3 + x^4 + x^5 + 3*x^6 + x^7 + 2*x^8 + 2*x^9 + ...
G.f. = 1/q - q^23 + 2*q^47 + q^71 + q^95 + q^119 + 3*q^143 + q^167 + ...

Mathematica

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

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^10 / (eta(x^2 + A)^3 * eta(x^3 + A)^4 * eta(x^12 + A)^4), n))};

Crossrefs

Cf. A259538.

Sequence in context: A285728 A157925 A099244 * A014671 A029365 A204178

Adjacent sequences: A260218 A260219 A260220 * A260222 A260223 A260224

Keyword

sign

Author

Michael Somos, Jul 19 2015

Status

approved