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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. 4

%I

%S 3,-42,393,-3240,24999,-184740,1325679,-9312408,64364025,-439225086,

%T 2966629452,-19868187384,132119675241,-873278632080,5742216378024,

%U -37587341460600,245063740036086,-1592173816624290,10311978807488160

%N 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 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%C Coefficients in a certain q-series associated with a failed attempt to explain a mysterious entry in a Ramanujan notebook.

%D B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 179, Eq. 13.23.

%H Vaclav Kotesovec, <a href="/A051273/b051273.txt">Table of n, a(n) for n = 0..1250</a>

%F Expansion of 3*(eta(q)*eta(q^3))^2/(theta[A_2](q)^2*q^(1/3)) in powers of q.

%F 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

%e G.f. = 3 - 42*x + 393*x^2 - 3240*x^3 + 24999*x^4 - 184740*x^5 + ...

%e G.f. = 3*q - 42*q^4 + 393*q^7 - 3240*q^10 + 24999*x^13 - 184740*q^16 + ...

%t 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-Fran├žois Alcover_, Nov 06 2015, adapted from PARI *)

%t 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 (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 Cf. A004016, A058092, A258941, A258942.

%K sign

%O 0,1

%A _N. J. A. Sloane_

%E Corrected and extended by Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 15 2000

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Last modified July 18 07:10 EDT 2019. Contains 325134 sequences. (Running on oeis4.)