A197355 - OEIS

%I #33 May 22 2025 20:57:24

%S 1,8,66,560,4876,43488,396664,3695168,35114640,340039808,3352381984,

%T 33619852032,342711219904,3548566208000,37299021381504,

%U 397752024876032,4300986840453376,47135959519660032,523321228732695040,5883464372569321472,66953921672396983296

%N Special values of Hermite polynomials.

%H Vincenzo Librandi, <a href="/A197355/b197355.txt">Table of n, a(n) for n = 0..200</a>

%F In Maple notation, a(n)=I^n*HermiteH(n,-4*I), n=0,1... .

%F E.g.f.: exp(8*x+x^2).

%F Recurrence: a(n) = 8*a(n-1) + 2*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012

%F a(n) ~ 2^(n/2-1/2)*exp(4*sqrt(2*n)-n/2-8)*n^(n/2)*(1+19/3*sqrt(2)/sqrt(n)). - _Vaclav Kotesovec_, Oct 20 2012

%t CoefficientList[Series[E^(8*x+x^2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 20 2012 *)

%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( 8*x + x^2 + x*O(x^n) ), n))} /* _Michael Somos_, Oct 29 2011 */

%Y Cf. A079949 (H(n,-3*I)).

%K nonn

%O 0,2

%A _Karol A. Penson_, Oct 17 2011