A393810 Expansion of 2F1(1/6, 1/2; 1/2; 27*x/(1-4*x)^3)^2.

1, 9, 270, 8154, 259209, 8529921, 287329140, 9841383288, 341309814858, 11951985351834, 421773730944084, 14977884769617708, 534677344017297858, 19171218712802918706, 689996917303356051792, 24915114884708594195832, 902230812743891520644937, 32753980020160125318149649

Offset

0,2

Comments

Integer sequence arising from the symmetric square of a Gauss hypergeometric function composed with a degree-3 Belyi map phi(x) = 27*x/(1-4*x)^3. Not an Apery-like sequence: no order-2 polynomial recurrence was found up to degree 12.

Links

Table of n, a(n) for n=0..17.

Alex Shvets, Integer sequences from Sym^2(2F1) Belyi pullback scan, Zenodo, 2026.

Formula

a(n) = [x^n] 2F1(1/6, 1/2; 1/2; 27*x/(1-4*x)^3)^2.

Recurrence: (3*n+4)*(n+3)*a(n+3) - 3*(39*n^2+143*n+124)*a(n+2) + 144*(n+1)*(n+2)*a(n+1) - 64*n*(3*n+7)*a(n) = 0.

a(n) ~ cosh(arcsinh(1)/3)^(1/3) * (1 + 2*cosh(arccosh(3)/3))^(3*n) / (2^(1/6) * Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Mar 30 2026

Example

For n=0 the recurrence gives (3*0+4)*(0+3)*a(3) - 3*(39*0^2+143*0+124)*a(2) + 144*(0+1)*(0+2)*a(1) - 64*0*(3*0+7)*a(0) = 0, i.e., 12*a(3) - 372*270 + 288*9 = 0, so a(3) = 8154.

Mathematica

CoefficientList[Series[Hypergeometric2F1[1/6, 1/2, 1/2, 27x/(1-4x)^3]^2, {x, 0, 17}], x] (* Stefano Spezia, Mar 29 2026 *)

Crossrefs

Cf. A394679, A394680.

Sequence in context: A168600 A088672 A300169 * A157571 A202689 A364438

Adjacent sequences: A393806 A393807 A393809 * A393811 A393812 A393813

Keyword

nonn

Author

Alex Shvets, Mar 28 2026

Extensions

Corrected and extended by Stefano Spezia, Mar 29 2026

Status

approved