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A277381 a(n) = exp(-1) * Sum_{k>=0} H_n(k)/k!, where H_n(x) is n-th Hermite polynomial.

%I A277381 #23 Feb 16 2025 08:33:36
%S A277381 1,2,6,28,156,984,7112,57488,508688,4887840,50639200,561416640,
%T A277381 6621963712,82719611264,1089925195904,15094182365440,219035852333312,
%U A277381 3321573144437248,52511336993334784,863594855161338880,14746060593172585472,260971042697438779392
%N A277381 a(n) = exp(-1) * Sum_{k>=0} H_n(k)/k!, where H_n(x) is n-th Hermite polynomial.
%H A277381 G. C. Greubel, <a href="/A277381/b277381.txt">Table of n, a(n) for n = 0..500</a>
%H A277381 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>
%H A277381 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>
%F A277381 E.g.f.: exp(exp(2*x)-x^2-1).
%F A277381 a(n) ~ 2^n * n^n * exp(n/LambertW(n) - LambertW(n)^2 / 4 - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^n). - _Vaclav Kotesovec_, Jun 29 2022
%p A277381 g := exp(exp(2*x)-x^2-1): ser := series(g,x,32):
%p A277381 seq(n!*coeff(ser,x,n), n=0..21); # _Peter Luschny_, Oct 19 2016
%t A277381 Table[Sum[HermiteH[n, k]/k!, {k, 0, Infinity}]/E, {n, 0, 20}] (* or *)
%t A277381 Table[SeriesCoefficient[Exp[Exp[2 x] - x^2 - 1], {x, 0, n}] n!, {n, 0, 20}]
%t A277381 With[{nmax = 50}, CoefficientList[Series[Exp[Exp[2*x] - x^2 - 1], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Jul 12 2018 *)
%t A277381 Table[Sum[Binomial[n, k] * 2^k * BellB[k] * HermiteH[n-k, 0], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 13 2018 *)
%o A277381 (PARI) x='x+O('x^50); Vec(serlaplace(exp(exp(2*x)-x^2-1))) \\ _G. C. Greubel_, Jul 12 2018
%o A277381 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(2*x) -x^2 -1))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Jul 12 2018
%Y A277381 Cf. A277380, A316778.
%K A277381 nonn
%O A277381 0,2
%A A277381 _Vladimir Reshetnikov_, Oct 11 2016