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
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..500
- Eric Weisstein's World of Mathematics, Hermite Polynomial
- Wikipedia, Hermite polynomials
Programs
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Magma
m:=50; R
:=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 -
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
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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 *) -
PARI
x='x+O('x^50); Vec(serlaplace(exp(exp(2*x)-x^2-1))) \\ G. C. Greubel, Jul 12 2018
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