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%I #9 May 11 2021 03:09:07
%S 2,7,136,30096,128231424,15917683507200,81063451589345280000,
%T 22675515428700722036736000000,
%U 449302248871829829537656890982400000000,790103237429135552913731284331032467210240000000000
%N a(n)= det[A000522(i+j+1)], i,j=0...n, is the Hankel determinant of order n+1 of the arrangements numbers, s. A000522; a(n) = product( (p!)^2,p=0..n )*(n+1)!*LaguerreL(n+1,0,-1), n=0,1..., where LaguerreL(n,lambda,x) are generalized Laguerre polynomials; a(n)=A055209(n)*A002720(n+1);.
%F a(n) ~ 2^(n+1/2) * Pi^(n+1) * n^(n^2 + 3*n + 25/12) / (A^2 * exp(3*n^2/2 + 3*n - 2*sqrt(n) + 1/3)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, May 11 2021
%t a[n_] := Det[Table[E*Gamma[i+j+2, 1] // FunctionExpand, {i, 0, n}, {j, 0, n}]];
%t Table[a[n], {n, 0, 9}] (* _Jean-François Alcover_, May 23 2016 *)
%t Table[BarnesG[n+2]^2 * (n+1)! * LaguerreL[n+1, 0, -1], {n, 0, 12}] (* _Vaclav Kotesovec_, May 11 2021 *)
%Y Cf. A000522, A055209, A002720.
%K nonn
%O 0,1
%A _Karol A. Penson_, Dec 16 2004
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