|
%I #15 Aug 14 2018 21:05:26
%S 1,6,0,9,24,0,27,84,0,82,222,0,207,558,0,486,1260,0,1055,2724,0,2205,
%T 5550,0,4374,10944,0,8427,20778,0,15696,38448,0,28539,69228,0,50630,
%U 122118,0,88119,210966,0,150417,358356,0,252727,598650,0,418068,986022
%N Expansion of q^(3/8) * a(q) / eta(q^3)^3 in powers of q where a() is a cubic AGM function.
%H G. C. Greubel, <a href="/A187429/b187429.txt">Table of n, a(n) for n = 0..2500</a>
%H J. M. Borwein and P. B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
%F Expansion of q^(3/8) * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^4 in powers of q.
%F 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.
%F a(3*n) = A053762(n). a(3*n+ 1) = 6 * A187428(n). a(3*n + 2) = 0.
%e G.f. = 1 + 6*x + 9*x^3 + 24*x^4 + 27*x^6 + 84*x^7 + 82*x^9 + 222*x^10 + ...
%e 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 + ...
%t 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 *)
%o (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))}
%Y Cf. A053762, A187428.
%K nonn
%O 0,2
%A _Michael Somos_, Mar 09 2011
|
