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a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} 1 / ((n-2*k)! * 4^k * k!).
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%I #36 Mar 06 2022 08:34:41

%S 1,1,3,15,114,1170,15570,256410,5103000,119773080,3264445800,

%T 101784097800,3591396824400,141958074258000,6236035482877200,

%U 302218901402418000,16060366291617648000,930654556409161584000,58524794739862410960000,3976525824684785163792000

%N a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} 1 / ((n-2*k)! * 4^k * k!).

%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + x^2 / 4 ).

%F a(n) = n! * Sum_{k=0..n} Stirling1(n,k) * Bell(k) / 2^(n-k).

%F D-finite with recurrence a(0) = a(1) = 1; a(n) = n * a(n-1) + n * (n-1)^2 * a(n-2) / 2.

%F a(n) ~ sqrt(Pi) * n^((3*n + 1)/2) / (2^(n/2) * exp((3*n + 1)/2 - sqrt(2*n))). - _Vaclav Kotesovec_, Jul 17 2021

%t Table[(n!)^2 Sum[1/((n - 2 k)! 4^k k!), {k, 0, Floor[n/2]}], {n, 0, 19}]

%t nmax = 19; CoefficientList[Series[Exp[x + x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2

%o (PARI) a(n) = (n!)^2 * sum(k=0, n\2, 1/((n-2*k)!*4^k*k!)); \\ _Michel Marcus_, Jul 17 2021

%Y Cf. A000085, A000898, A023998, A080599, A239840.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Jul 16 2021