A328785 Expansion of q^(-1/3) * (1/3) * b(q)*c(q)/a(q)^2 in powers of q where a(), b(), c() are cubic AGM functions.

1, -14, 131, -1080, 8333, -61580, 441893, -3104136, 21454675, -146408362, 988876484, -6622729128, 44039891747, -291092877360, 1914072126008, -12529113820200, 81687913345362, -530724605541430, 3437326269162720, -22199991545327616, 143016156285625823

Offset

0,2

Comments

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

Convolution square of A258941. Convolution inverse of A058092 with more information there.

References

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1998.

Links

Table of n, a(n) for n=0..20.

Formula

Coefficients of 1/3 of power series in equation (13.23), page 179, [Berndt 1998]

A051273(n) = 3*a(n).

a(n) ~ (-1)^n * 64 * Pi^5 * n * exp(Pi*(3*n+1)/3^(3/2)) / Gamma(1/6)^6. - Vaclav Kotesovec, Nov 08 2019

Example

G.f. = 1 - 14*x + 131*x^2 - 1080*x^3 + 8333*x^4 - 61580*x^5 + ...
G.f. = q - 14*q^4 + 131*q^7 - 1080*q^10 + 8333*q^13 - 61580*q^16 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[x]^2 QPochhammer[x^3]^4 / (QPochhammer[x]^3 + 9 x QPochhammer[x^9]^3)^2, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); A = ((eta(x + A) * eta(x^3 + A)) / (eta(x^2 + A) * eta(x^6 + A)))^2; polcoeff( 1 / (A + x * 16/A^2), n))};

Crossrefs

Cf. A058092, A258941, A051273.

Sequence in context: A358168 A026936 A021114 * A113976 A022738 A017269

Adjacent sequences: A328782 A328783 A328784 * A328786 A328787 A328788

Keyword

sign

Author

Michael Somos, Nov 02 2019

Status

approved