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 A352149 #10 Mar 09 2022 10:40:45
%S A352149 0,1,5,20,90,499,3395,27474,256984,2720169,32080501,416574212,
%T A352149 5900292266,90461885331,1491788697451,26318520300986,494449968500832,
%U A352149 9852544385880961,207497251731808341,4604297325494524516,107348917822006139114,2623224641748615607715,67035139167875735937219
%N A352149 a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * (n-k-1)!.
%F A352149 Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) * (Ei(x) - log(x) - gamma).
%F A352149 From _Vaclav Kotesovec_, Mar 06 2022: (Start)
%F A352149 Recurrence: n*(n^3 - 12*n^2 + 46*n - 50)*a(n) = (2*n^5 - 24*n^4 + 95*n^3 - 123*n^2 + 37*n - 2)*a(n-1) - (n^6 - 12*n^5 + 51*n^4 - 95*n^3 + 121*n^2 - 139*n + 74)*a(n-2) + (n-2)*(2*n^5 - 27*n^4 + 140*n^3 - 344*n^2 + 409*n - 173)*a(n-3) - (n-3)^2*(n-2)*(n^3 - 9*n^2 + 25*n - 15)*a(n-4).
%F A352149 a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n - 3/4) / sqrt(2). (End)
%t A352149 Table[Sum[Binomial[n, k]^2 (n - k - 1)!, {k, 0, n - 1}], {n, 0, 22}]
%t A352149 nmax = 22; Assuming[x > 0, CoefficientList[Series[BesselI[0, 2 Sqrt[x]] (ExpIntegralEi[x] - Log[x] - EulerGamma), {x, 0, nmax}], x]] Range[0, nmax]!^2
%o A352149 (PARI) a(n) = sum(k=0, n-1, binomial(n,k)^2 * (n-k-1)!); \\ _Michel Marcus_, Mar 06 2022
%Y A352149 Cf. A002104, A002720, A352150.
%K A352149 nonn
%O A352149 0,3
%A A352149 _Ilya Gutkovskiy_, Mar 06 2022