A097388 2n-th derivative of the Gaussian exp(-x^2) evaluated at x=0.

1, -2, 12, -120, 1680, -30240, 665280, -17297280, 518918400, -17643225600, 670442572800, -28158588057600, 1295295050649600, -64764752532480000, 3497296636753920000, -202843204931727360000, 12576278705767096320000, -830034394580628357120000, 58102407620643984998400000

Offset: 0

Author

Michael Somos, Aug 12 2004

Keywords

• sign

Comments

H_n(0) := n-th derivative of exp(-x^2) at x=0. H_(2*n + 1)(0) = 0 if n >= 0. - Michael Somos, Jan 24 2014

Self-convolution gives A076729(n)*(-1)^n. - Vladimir Reshetnikov, Oct 11 2016

Examples

			exp(-x^2) = 1 - x^2 + x^4/4 - x^6/6 + ..., (d/dx)^4 exp(-x^2) = 12 - 60x^2 + ... so a(2)=12.
G.f. = 1 - 2*x + 12*x^2 - 120*x^3 + 1680*x^4 - 30240*x^5 + 665280*x^6 + ...
		

Crossrefs

Cf. A001813, A076729.

Programs

Maple

A097388 := n -> (-2)^n*n!*JacobiP(n, -1/2, -n, 3):
seq(simplify( A097388(n)), n = 0..18);  # Peter Luschny, Jan 22 2025

Mathematica

a[ n_] := If[ n < 0, 0, HermiteH[ 2 n, 0]]; (* Michael Somos, Jan 24 2014 *)

PARI

{a(n) = if( n<0, 0, (-1)^n * (2*n)! / n!)};

Formula

E.g.f.: Sum_{k>=0} a(k) * x^(2*k) / (2*k)! = exp(-x^2).

a(n) = (-1)^n *(2*n)! / n!.

G.f.: 1/U(0) where U(k) = 1 + x*(2*k+2)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Nov 14 2012

G.f.: 1/Q(0), where Q(k) = 1 - x*(8*k+2) + x*(8*k+4)/(1 - x*(8*k+6) + x*(8*k+8)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 18 2013

a(n) = (-1)^n * A001813(n).

G.f. = 1 / (1 + 2*x / (1 + 4*x / (1 + 6*x / ... ))). - Michael Somos, Jan 24 2014

a(n) ~ (-1)^n*2^(2*n+1/2)*n^n/exp(n). - Ilya Gutkovskiy, Oct 11 2016

Sum_{n>=0} 1/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)). - Amiram Eldar, Nov 12 2020

From Nikolaos Pantelidis, Jan 08 2021: (Start)

G.f.: 1/G(1), where G(n) = 1+(8*n-6)*x-8*n*(2*n-1)*x^2/G(n+1); (Jacobi continued fraction)

G.f.: 1/(1 + 2*x - 8*x^2/(1 + 10*x - 48*x^2/(1 + 18*x - 120*x^2/(1 + 26*x - 224*x^2/(1 + 34*x - 360*x^2/(1+ 42*x - 448*x^2/(1+ 50*x - 648*x^2/( ...)))))))) (Jacobi continued fraction). (End)

a(n) = (-2)^n*n!*JacobiP(n, -1/2, -n, 3). - Peter Luschny, Jan 22 2025