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Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(exp(x) - 1)).
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%I #9 Jul 28 2020 11:15:16

%S 1,2,9,67,725,10616,200767,4740149,136113217,4656324934,186642121061,

%T 8647446227487,457854954921949,27435354945248732,1844986431192663683,

%U 138229607701444447561,11464234006789370705537,1046538415206891196153834,104623195637603009050593697

%N Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(exp(x) - 1)).

%F a(n) = n! * Sum_{k=0..n} Stirling2(n+1,k+1) / k!.

%t nmax = 18; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!^2

%t Table[n! Sum[StirlingS2[n + 1, k + 1]/k!, {k, 0, n}], {n, 0, 18}]

%o (PARI) a(n) = n! * sum(k=0, n, stirling(n+1,k+1,2) / k!); \\ _Michel Marcus_, Jul 27 2020

%Y Cf. A000110, A002720, A119392, A336589.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jul 26 2020