A202394 Expansion of f(-x)^3 + 9 * x * f(-x^9)^3 in powers of x where f() is a Ramanujan theta function.

1, 6, 0, 5, 0, 0, -7, 0, 0, 0, -18, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0, -42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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^3) * a(x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.

Expansion of q^(-1/8) * (eta(q)^3 + 9 * eta(q^9)^3) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 41472^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A116916.

G.f.: Sum_{k} -(-1)^k * (6*k - 1) * x^(3*k*(3*k - 1)/2) + Sum_{k>0} -(-1)^k * 6 * (2*k - 1) * x^(9*k*(k - 1)/2 + 1).

a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(3*n) = A116916(n). a(9*n + 1) = 6 * A010816(n). a(25*n + 3) = 5 * a(n).

a(n) nonzero if and only if n is a triangular number.

Example

G.f. = 1 + 6*x + 5*x^3 - 7*x^6 - 18*x^10 - 11*x^15 + 13*x^21 + 30*x^28 + ...
G.f. = q + 6*q^9 + 5*q^25 - 7*q^49 - 18*q^81 - 11*q^121 + 13*q^169 + 30*q^225 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3, {x, 0, n}]; (* Michael Somos, May 26 2014 *)

Prog

(PARI) {a(n) = local(m); if( issquare(8*n + 1, &m), (-1)^(m \ 6) * m * ((m%3 == 0) + 1), 0)};
(PARI) {a(n) = local(A); if( n<0, 0,  A = x * O(x^n); polcoeff( eta(x + A)^3 + 9 * x * eta(x^9 + A)^3, n))};

Crossrefs

Cf. A010816, A116916.

Sequence in context: A113024 A112280 A204850 * A202954 A218850 A021627

Adjacent sequences: A202391 A202392 A202393 * A202395 A202396 A202397

Keyword

sign

Author

Michael Somos, Dec 18 2011

Status

approved