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

1, 1, 1, 5, 25, 97, 489, 3285, 22481, 160737, 1293041, 11348933, 105136937, 1033279873, 10808289561, 119401994709, 1385242479137, 16846680046657, 214333419288161, 2844927602028549, 39305588104667321, 564208058072724257, 8400178767847987401, 129509650839484638037

Offset: 0

Author

Vaclav Kotesovec, Jul 13 2018

Keywords

• nonn

Comments

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A277380, A277381.

Programs

Mathematica

Table[Exp[-1/2]*Sum[HermiteH[n, k]/k!/2^k, {k, 0, Infinity}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[Exp[2*x]/2 - x^2 - 1/2], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k, 1/2] * HermiteH[n-k, 0], {k, 0, n}], {n, 0, 20}]

Formula

E.g.f.: exp(exp(2*x)/2 - x^2 - 1/2).

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