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

0,4

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

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

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}]

Crossrefs

Cf. A277380, A277381.

Sequence in context: A273747 A201841 A146830 * A255612 A022729 A098111

Adjacent sequences: A316775 A316776 A316777 * A316779 A316780 A316781

Keyword

nonn

Author

Vaclav Kotesovec, Jul 13 2018

Status

approved