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).
1, 96, 14944, 2743296, 547115616, 114691716096, 24855999978496, 5516395226824704, 1246310097807086176, 285511424277840331776, 66136775263705972306944, 15459962390271174936920064, 3641349843333453310791883776, 863175698505287814277639471104, 205741271729612742942836920909824
Offset: 0
Keywords
Links
- Li Lai, Johannes Sprang, and Wadim Zudilin, A note on the irrationality of zeta_2(5), arXiv:2505.05005 [math.NT], 2025. See pp. 2, 10.
Programs
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Mathematica
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]
Formula
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).
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).