A394679 - OEIS

%I #30 Apr 11 2026 17:46:45

%S 1,3,17,95,537,3059,17513,100607,579377,3342627,19311873,111696991,

%T 646614345,3745984787,21714513369,125937787135,730719993441,

%U 4241392619843,24626744856305,143030575690079,830920303820921,4828209352231731,28060704773688393,163112737069681663

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

%C Integer sequence arising from the symmetric square of a Gauss hypergeometric function composed with a degree-2 rational map phi(x) = 4*x/(1-x)^2. 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(1/4, 3/4; 1/2; 4*x/(1-x)^2)^2.

%F Recurrence: (2*n+3)*(n+4)*a(n+4) - 6*(4*n^2+19*n+20)*a(n+3) + 4*(19*n^2+76*n+75)*a(n+2) - 6*(4*n^2+13*n+8)*a(n+1) + n*(2*n+5)*a(n) = 0.

%F a(n) ~ 2^(-3/2) * (1 + sqrt(2))^(2*n) * (1 + 2^(1/4)/sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 30 2026

%e For n=0 the recurrence reads 3*4*a(4) - 6*20*a(3) + 4*75*a(2) - 6*8*a(1) + 0*a(0) = 0, i.e., 12*a(4) = 120*95 - 300*17 + 48*3 = 11400 - 5100 + 144 = 6444, so a(4) = 6444/12 = 537.

%t CoefficientList[Series[Hypergeometric2F1[1/4, 3/4, 1/2, 4x/(1-x)^2]^2,{x,0,23}],x] (* _Stefano Spezia_, Mar 29 2026 *)

%Y Cf. A393810, A394680.

%K nonn

%O 0,2

%A _Alex Shvets_, Mar 28 2026

%E a(20)-a(23) from _Stefano Spezia_, Mar 29 2026