A394680 - OEIS

%I #37 Apr 13 2026 18:48:24

%S 1,8,208,5376,148480,4317184,130351104,4049600512,128634060800,

%T 4159180898304,136443125039104,4530275132375040,151953149056253952,

%U 5141222724013654016,175260243066041139200,6013762480240985112576,207546103051546813005824,7199550144452424240201728

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

%C 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.

%H Alex Shvets, <a href="https://doi.org/10.5281/zenodo.19387246">Integer sequences from Sym^2(2F1) Belyi pullback scan</a>, Zenodo, 2026.

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

%F 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.

%F 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

%e 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.

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

%Y Cf. A393810, A394679.

%K nonn

%O 0,2

%A _Alex Shvets_, Mar 28 2026

%E Corrected and extended by _Stefano Spezia_, Mar 29 2026