A394680 Expansion of 2F1(1/3, 2/3; 3/2; 27*x/(1-4*x)^3)^2.

1, 8, 208, 5376, 148480, 4317184, 130351104, 4049600512, 128634060800, 4159180898304, 136443125039104, 4530275132375040, 151953149056253952, 5141222724013654016, 175260243066041139200, 6013762480240985112576, 207546103051546813005824, 7199550144452424240201728

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 with 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(1/3, 2/3; 3/2; 27*x/(1-4*x)^3)^2.

Recurrence: (n+4)*(2*n+7)*(3*n-1)*(3*n+4)*(3*n+5)*(9*n^2+21*n-2)*a(n+3) - 6*n*(3159*n^6+34749*n^5+146529*n^4+288135*n^3+236736*n^2+5556*n-65264)*a(n+2) + 48*(n-1)*(3*n+8)*(162*n^5+1269*n^4+3600*n^3+5019*n^2+4238*n+1960)*a(n+1) - 128*(n-2)*(n-1)*(3*n+5)*(3*n+7)*(3*n+8)*(9*n^2+39*n+28)*a(n) = 0.

a(n) ~ 3^(3/2) * sqrt(2*cosh(arccosh(3)/3) - 1) * ((2*cosh(arccosh(3)/3) + 1)^(3*n) / (4*sqrt(Pi)*n^(3/2))). - Vaclav Kotesovec, Mar 30 2026

Example

For n=1 the recurrence gives 141120*a(4) - 3897600*a(3) = 0 (the last two terms vanish since n-1=0), so a(4) = 3897600*5376/141120 = 148480.

Mathematica

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

Crossrefs

Cf. A393810, A394679.

Sequence in context: A090962 A359281 A359452 * A330287 A279663 A294970

Adjacent sequences: A394677 A394678 A394679 * A394681 A394682 A394683

Keyword

nonn

Author

Alex Shvets, Mar 28 2026

Extensions

Corrected and extended by Stefano Spezia, Mar 29 2026

Status

approved