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 A382779 #13 May 22 2025 20:57:18
%S A382779 1,96,14944,2743296,547115616,114691716096,24855999978496,
%T A382779 5516395226824704,1246310097807086176,285511424277840331776,
%U A382779 66136775263705972306944,15459962390271174936920064,3641349843333453310791883776,863175698505287814277639471104,205741271729612742942836920909824
%N A382779 a(n) = Sum_{0<=i<=k<=n} 2^(4*(n-k)) * binomial(2*i,i)^2 * binomial(2*n-2*i,n-i) * binomial(2*k-2*i,k-i) * binomial(2*k,k)^2 * binomial(2*n-2*k,n-k).
%H A382779 Li Lai, Johannes Sprang, and Wadim Zudilin, <a href="https://arxiv.org/abs/2505.05005">A note on the irrationality of zeta_2(5)</a>, arXiv:2505.05005 [math.NT], 2025. See pp. 2, 10.
%F A382779 Recurrence: (n + 1)^5 * a(n+1) - 32 * (2*n + 1) * (8*n^4 + 16*n^3 + 20*n^2 + 12*n + 3) * a(n) + 2^16 * n^5 * a(n-1) = 0 (see Lai et al., p. 2).
%F A382779 a(n) = Sum_{k=0..n} 2^(4*(n-k)) * binomial(2*k,k)^3 * binomial(2*n,n) * binomial(2*n-2*k,n-k) * hypergeom([1/2, 1/2, -k, -n], [1, 1/2-k, 1/2-n], 1).
%t A382779 a[n_]:=Sum[Sum[2^(4(n-k))Binomial[2i,i]^2Binomial[2n-2i,n-i]Binomial[2k-2i,k-i]Binomial[2k,k]^2Binomial[2n-2k,n-k],{i,0,n}],{k,0,n}]; Array[a,15,0]
%Y A382779 Cf. A000984, A001025, A002894, A106187.
%K A382779 nonn
%O A382779 0,2
%A A382779 _Stefano Spezia_, May 11 2025