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A137685 Expansion of phi(-q^3) / f(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions. 1
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 (list; graph; refs; listen; history; text; internal format)
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).

M. 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: A171238 A096541 A226015 * A169826 A093065 A301596

Adjacent sequences:  A137682 A137683 A137684 * A137686 A137687 A137688

KEYWORD

nonn

AUTHOR

Michael Somos, Feb 05 2008

STATUS

approved

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Last modified March 22 12:43 EDT 2019. Contains 321421 sequences. (Running on oeis4.)