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%I #12 Mar 17 2024 11:36:30
%S 1,2,22,438,12824,496370,23914512,1379269094,92667551104,
%T 7108231236066,612974464428800,58702772664490262,6181602019316333568,
%U 709911177607125141362,88301595129435811723264,11825985945777638231211750,1696696168760520436580974592,259624546758869333450285984066
%N a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * n^(n-k).
%F a(n) = n!^3 * [x^n] BesselI(0,2*sqrt(x)) * Sum_{k>=0} n^k * x^k / k!^3.
%F a(n) ~ c * n^(n - 1/2) / (exp(r*n) * r^(2*n)), where r = (2 - 5*(2/(3*sqrt(69)-11))^(1/3) + ((3*sqrt(69)-11)/2)^(1/3))/3 = 0.430159709001946734... is the real root of the equation r^2 = (1-r)^3 and c = sqrt(138 + 2^(2/3)*(69*(8901 - 223*sqrt(69)))^(1/3) + 2^(2/3)*(69*(8901 + 223*sqrt(69)))^(1/3))/(2*sqrt(69*Pi)) = 0.684738330749970434111338151096549475398274404060139170789278633219363118... - _Vaclav Kotesovec_, Jul 01 2022, updated Mar 17 2024
%t Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k]^3 k! n^(n - k), {k, 0, n}], {n, 0, 17}]
%t Unprotect[Power]; 0^0 = 1; Table[n!^3 SeriesCoefficient[BesselI[0, 2 Sqrt[x]] Sum[n^k x^k/k!^3, {k, 0, n}], {x, 0, n}], {n, 0, 17}]
%o (PARI) a(n) = sum(k=0, n, binomial(n,k)^3 * k! * n^(n-k)); \\ _Michel Marcus_, Jun 12 2022
%Y Cf. A000172, A063170, A216831, A241247, A274246, A277373, A277386, A354944.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 12 2022
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