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A051273 Expansion of q^(-1/3) * b(q) * c(q) / a(q)^2 in powers of q where a(), b(), c() are cubic AGM theta functions.

%I A051273 #31 Nov 08 2019 02:50:24
%S A051273 3,-42,393,-3240,24999,-184740,1325679,-9312408,64364025,-439225086,
%T A051273 2966629452,-19868187384,132119675241,-873278632080,5742216378024,
%U A051273 -37587341460600,245063740036086,-1592173816624290,10311978807488160
%N A051273 Expansion of q^(-1/3) * b(q) * c(q) / a(q)^2 in powers of q where a(), b(), c() are cubic AGM theta functions.
%C A051273 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A051273 Coefficients in a certain q-series associated with a failed attempt to explain a mysterious entry in a Ramanujan notebook.
%D A051273 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 179, Eq. 13.23.
%H A051273 Vaclav Kotesovec, <a href="/A051273/b051273.txt">Table of n, a(n) for n = 0..1250</a>
%F A051273 Expansion of 3*(eta(q)*eta(q^3))^2/(theta[A_2](q)^2*q^(1/3)) in powers of q.
%F A051273 a(n) ~ (-1)^n * c * n * exp(Pi*n/sqrt(3)), where c = 3 * A258942^2 = 192 * exp(Pi/(3*sqrt(3))) * Pi^5 / Gamma(1/6)^6 = 3.6159115405362166049256277... . - _Vaclav Kotesovec_, Nov 07 2015, updated Nov 14 2015
%F A051273 a(n) = 3*A328785(n). - _Michael Somos_, Nov 02 2019
%e A051273 G.f. = 3 - 42*x + 393*x^2 - 3240*x^3 + 24999*x^4 - 184740*x^5 + ...
%e A051273 G.f. = 3*q - 42*q^4 + 393*q^7 - 3240*q^10 + 24999*x^13 - 184740*q^16 + ...
%t A051273 a[0] = 3; a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[3*(QPochhammer[ x + A]*(QPochhammer[x^3 + A]^2/(QPochhammer[x + A]^3 + 9*x * QPochhammer[ x^9 + A]^3)))^2, n]]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 06 2015, adapted from PARI *)
%t A051273 CoefficientList[Series[3 * (QPochhammer[x,x] * QPochhammer[x^3,x^3]^2 / (QPochhammer[x,x]^3 + 9*x*QPochhammer[x^9,x^9]^3))^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Nov 07 2015 *)
%o A051273 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * (eta(x + A) * eta(x^3 + A)^2 / (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3))^2, n))}; /* _Michael Somos_, Aug 07 2006 */
%Y A051273 Cf. A004016, A058092, A258941, A258942, A328785.
%K A051273 sign
%O A051273 0,1
%A A051273 _N. J. A. Sloane_
%E A051273 Corrected and extended by Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 15 2000