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%I #12 Jun 19 2022 08:29:20
%S 1,4,35,438,6873,127488,2703447,64121130,1674999009,47638235484,
%T 1461975938379,48068355965886,1683311251028265,62477888170824792,
%U 2447583053876363727,100842325515959413842,4356021203508275392833,196739133595421931988020,9268144156277932321747251
%N a(n) = Sum_{k=0..n} binomial(n, k)^3 * 3^(n-k) * k!.
%F Recurrence: n*(8*n - 23)*a(n) = 3*(8*n^3 - 15*n^2 - 30*n + 17)*a(n-1) - (n-1)*(24*n^3 - 261*n^2 + 770*n - 666)*a(n-2) + (n-2)^3*(n-1)*(8*n - 15)*a(n-3).
%F a(n) ~ n^(n - 1/6) * exp(3*3^(1/3)*n^(2/3) - 3^(2/3)*n^(1/3) - n +1) / (3^(5/6)*sqrt(2*Pi)) * (1 + 19/(6*3^(2/3)*n^(1/3)) + 1193/(1080*3^(1/3) * n^(2/3))).
%F Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} 3^n * x^n / n!^3. - _Ilya Gutkovskiy_, Jun 19 2022
%t Table[Sum[Binomial[n, k]^3 * 3^(n-k) * k!, {k, 0, n}], {n, 0, 20}]
%Y Cf. A000172, A277382, A241247, A274246.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Oct 12 2016
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