This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381199 #14 May 29 2025 02:21:29
%S A381199 1,36,6300,1718640,575675100,216636756336,87874675224336,
%T A381199 37563969509352000,16692217815436148700,7642084994921759382000,
%U A381199 3582530520581922083974800,1712083670316898167464884800,831357643152788660610464490000,409154554816583487288034143528000,203690783136217174743485058666840000
%N A381199 a(n) = (4*n)!/((n!)^2*(2*n)!)*Sum_{k=0..n} binomial(n,k)^2*binomial(2*k,k).
%H A381199 S. Hassani, J.-M. Maillard, and N. Zenine, <a href="https://arxiv.org/abs/2502.05543">On the diagonals of rational functions: the minimal number of variables (unabridged version)</a>, arXiv:2502.05543 [math-ph], 2025. See p. 24.
%F A381199 a(n) = (4*n)!*hypergeom([1/2, -n, -n], [1, 1], 4)/((n!)^2*(2*n)!).
%F A381199 D-finite with recurrence n^4*a(n) -4*(4*n-1)*(4*n-3)*(10*n^2-10*n+3)*a(n-1) +144*(4*n-5)*(4*n-3)*(4*n-7)*(4*n-1)*a(n-2)=0. - _R. J. Mathar_, Feb 18 2025
%F A381199 a(n) ~ 2^(6*n - 1/2) * 3^(2*n + 3/2) / (4*Pi^2*n^2). - _Vaclav Kotesovec_, May 29 2025
%t A381199 a[n_]:=(4n)!/((n!)^2*(2n)!)*Sum[Binomial[n,k]^2Binomial[2k,k],{k,0,n}]; Array[a,15,0]
%Y A381199 Cf. A000897, A000984, A001044, A002893, A007318, A008459, A010050, A100733.
%K A381199 nonn
%O A381199 0,2
%A A381199 _Stefano Spezia_, Feb 16 2025