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A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + n)^n / (2^k * k!).

%I A337012 #4 Aug 11 2020 19:48:17
%S A337012 1,2,11,92,1025,14232,236403,4568720,100670529,2490511776,68341981051,
%T A337012 2059882505408,67645498798721,2403948686290816,91914992104815459,
%U A337012 3762299973887526144,164148252324092964993,7604537914425558921728,372812121514187124192875
%N A337012 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + n)^n / (2^k * k!).
%F A337012 a(n) = n! * [x^n] exp(n*x + (exp(2*x) - 1) / 2).
%F A337012 a(n) = Sum_{k=0..n} binomial(n,k) * n^(n-k) * A004211(k).
%t A337012 Table[n! SeriesCoefficient[Exp[n x + (Exp[2 x] - 1)/2], {x, 0, n}], {n, 0, 18}]
%t A337012 Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] n^(n - k) 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 18}]
%Y A337012 Cf. A004211, A007405, A134980, A337010, A337011.
%K A337012 nonn
%O A337012 0,2
%A A337012 _Ilya Gutkovskiy_, Aug 11 2020