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

1, 2, 6, 28, 156, 984, 7112, 57488, 508688, 4887840, 50639200, 561416640, 6621963712, 82719611264, 1089925195904, 15094182365440, 219035852333312, 3321573144437248, 52511336993334784, 863594855161338880, 14746060593172585472, 260971042697438779392

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Hermite Polynomial

Wikipedia, Hermite polynomials

Formula

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

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

Maple

g := exp(exp(2*x)-x^2-1): ser := series(g, x, 32):
seq(n!*coeff(ser, x, n), n=0..21); # Peter Luschny, Oct 19 2016

Mathematica

Table[Sum[HermiteH[n, k]/k!, {k, 0, Infinity}]/E, {n, 0, 20}] (* or *)
Table[SeriesCoefficient[Exp[Exp[2 x] - x^2 - 1], {x, 0, n}] n!, {n, 0, 20}]
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 *)
Table[Sum[Binomial[n, k] * 2^k * BellB[k] * HermiteH[n-k, 0], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 13 2018 *)

Prog

(PARI) x='x+O('x^50); Vec(serlaplace(exp(exp(2*x)-x^2-1))) \\ G. C. Greubel, Jul 12 2018
(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

Crossrefs

Cf. A277380, A316778.

Sequence in context: A295792 A004984 A326928 * A184695 A086633 A354311

Adjacent sequences: A277378 A277379 A277380 * A277382 A277383 A277384

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 11 2016

Status

approved