A187429 Expansion of q^(3/8) * a(q) / eta(q^3)^3 in powers of q where a() is a cubic AGM function.

1, 6, 0, 9, 24, 0, 27, 84, 0, 82, 222, 0, 207, 558, 0, 486, 1260, 0, 1055, 2724, 0, 2205, 5550, 0, 4374, 10944, 0, 8427, 20778, 0, 15696, 38448, 0, 28539, 69228, 0, 50630, 122118, 0, 88119, 210966, 0, 150417, 358356, 0, 252727, 598650, 0, 418068, 986022

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.

Formula

Expansion of q^(3/8) * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^4 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = (3/8)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A053762.

a(3*n) = A053762(n). a(3*n+ 1) = 6 * A187428(n). a(3*n + 2) = 0.

Example

G.f. = 1 + 6*x + 9*x^3 + 24*x^4 + 27*x^6 + 84*x^7 + 82*x^9 + 222*x^10 + ...
G.f. = q^-3 + 6*q^5 + 9*q^21 + 24*q^29 + 27*q^45 + 84*q^53 + 82*q^69 + 222*q^77 + ...

Mathematica

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x + A]^3 + 9*x*QPochhammer[x^9 + A]^3)/QPochhammer[x^3 + A]^4, {x, 0, n}]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 06 2015, adapted from PARI *)

Prog

(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) / eta(x^3 + A)^4, n))}

Crossrefs

Cf. A053762, A187428.

Sequence in context: A176403 A153314 A019622 * A083573 A117006 A294345

Adjacent sequences: A187426 A187427 A187428 * A187430 A187431 A187432

Keyword

nonn

Author

Michael Somos, Mar 09 2011

Status

approved