A134340 Expansion of psi(x)^3 * f(-x^3)^3 / chi(-x)^2 in powers of x where psi(), chi(), f() are Ramanujan theta functions.

1, 5, 12, 22, 35, 50, 70, 92, 117, 145, 170, 210, 250, 287, 330, 362, 425, 477, 532, 600, 626, 715, 782, 850, 925, 962, 1100, 1162, 1247, 1335, 1370, 1520, 1617, 1750, 1810, 1850, 2040, 2147, 2262, 2380, 2451, 2625, 2752, 2882, 3015, 3005, 3290, 3500, 3577

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..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 6 sequence [ 5, -3, 2, -3, 5, -6, ...].

-24 * a(n) = A103440(6*n + 5). 216 * a(n) = A109041(6*n + 5).

Example

G.f. = 1 + 5*x + 12*x^2 + 22*x^3 + 35*x^4 + 50*x^5 + 70*x^6 + 92*x^7 + 117*x^8 + ...
G.f. = q^5 + 5*q^11 + 12*q^17 + 22*q^23 + 35*q^29 + 50*q^35 + 70*q^41 + 94*q^47 + ...

Mathematica

a[ n_] := If[ n < 0, 0, (-1/24) DivisorSum[ 6 n + 5, #^2 KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Oct 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^2 QPochhammer[ x^3]^3 EllipticTheta[ 2, 0, x^(1/2)]^3 / (8 x^(3/8)), {x, 0, n}]; (* Michael Somos, Oct 25 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, n = 6*n + 5; sumdiv(n, d, d^2 * kronecker( -3, d)) / -24 )};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A)^3 / eta(x + A)^5, n))};

Crossrefs

Cf. A103440, A109041.

Sequence in context: A131976 A074376 A293502 * A000326 A022795 A025734

Adjacent sequences: A134337 A134338 A134339 * A134341 A134342 A134343

Keyword

nonn

Author

Michael Somos, Oct 21 2007

Status

approved