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

Author

Michael Somos, Mar 09 2011

Keywords

• nonn

Examples

			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 + ...
		

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.

Crossrefs

Cf. A053762, A187428.

Programs

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 *)

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))}

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.