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A316778 a(n) = exp(-1/2) * Sum_{k>=0} H_n(k) / (k!*2^k), where H_n(x) is n-th Hermite polynomial.

%I A316778 #13 Jun 29 2022 05:01:15
%S A316778 1,1,1,5,25,97,489,3285,22481,160737,1293041,11348933,105136937,
%T A316778 1033279873,10808289561,119401994709,1385242479137,16846680046657,
%U A316778 214333419288161,2844927602028549,39305588104667321,564208058072724257,8400178767847987401,129509650839484638037
%N A316778 a(n) = exp(-1/2) * Sum_{k>=0} H_n(k) / (k!*2^k), where H_n(x) is n-th Hermite polynomial.
%C A316778 In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + q*x^2 + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) + q*LambertW(n/m)^2 / b^2 - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - _Vaclav Kotesovec_, Jun 29 2022
%H A316778 Vaclav Kotesovec, <a href="/A316778/b316778.txt">Table of n, a(n) for n = 0..500</a>
%F A316778 E.g.f.: exp(exp(2*x)/2 - x^2 - 1/2).
%F A316778 a(n) ~ 2^n * n^n * exp(n/LambertW(2*n) - LambertW(2*n)^2 / 4 - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^n). - _Vaclav Kotesovec_, Jun 29 2022
%t A316778 Table[Exp[-1/2]*Sum[HermiteH[n, k]/k!/2^k, {k, 0, Infinity}], {n, 0, 20}]
%t A316778 nmax = 20; CoefficientList[Series[Exp[Exp[2*x]/2 - x^2 - 1/2], {x, 0, nmax}], x] * Range[0, nmax]!
%t A316778 Table[Sum[Binomial[n, k] * 2^k * BellB[k, 1/2] * HermiteH[n-k, 0], {k, 0, n}], {n, 0, 20}]
%Y A316778 Cf. A277380, A277381.
%K A316778 nonn
%O A316778 0,4
%A A316778 _Vaclav Kotesovec_, Jul 13 2018