A137685 Expansion of phi(-q^3) / f(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.

1, 2, 5, 8, 16, 26, 45, 70, 113, 170, 261, 382, 567, 812, 1171, 1646, 2322, 3212, 4448, 6066, 8272, 11142, 14992, 19970, 26561, 35032, 46117, 60280, 78631, 101946, 131888, 169724, 217937, 278548, 355237, 451178, 571799, 722002, 909744, 1142502, 1431889

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 71, Equ. (7.30). MR0858826 (88b:11063).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 6 sequence [ 2, 2, 0, 2, 2, 1, ...].

G.f.: Product_{k>0} (1 + x^k + x^(2*k)) / ( (1 - x^(2*k)) * (1 - x^k +x^(2*k)) ).

Example

G.f. = 1 + 2*q + 5*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 45*q^6 + 70*q^7 + 113*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] / QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Oct 04 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, sqrtint(n \ 3), 2 * (-1)^k * x^(3*k^2), 1 + A) / eta(x + A)^2, n))};
(PARI) {a(N) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))}; /* Michael Somos, Oct 04 2015 */
(PARI) q='q+O('q^99); Vec(eta(q^3)^2/(eta(q)^2*eta(q^6))) \\ Altug Alkan, Mar 30 2018

Crossrefs

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

Sequence in context: A096541 A226015 A328547 * A169826 A093065 A301596

Adjacent sequences: A137682 A137683 A137684 * A137686 A137687 A137688

Keyword

nonn

Author

Michael Somos, Feb 05 2008

Status

approved