%I #18 Jan 30 2020 04:34:53

%S 1,1,2,4,13,41,211,967,7274,44578,449551,3456641,43883797,405589549,

%T 6212792678,67022223556,1202604514141,14825243365517,304950638503279,

%U 4227716923246963,98067246206824406,1509933505953992386,38990856819985996927,660048542856323263589,18778057160849966289433

%N E.g.f.: exp(x-sqrt(1-x^2)+1).

%H Alois P. Heinz, <a href="/A259239/b259239.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) ~ (exp(2) + (-1)^n) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Jun 22 2015

%F a(n) = ((2*n-3)*a(n-1) + (n-1)*(n^2-5*n+5)*a(n-2) + 2*(n-1)*(n-2)*(n-3)*(a(n-4)-a(n-3)))/(n-2) for n >= 4. - _Alois P. Heinz_, Jan 30 2020

%p a:= proc(n) option remember; `if`(n<4, [1$2, 2, 4][n+1],

%p ((2*n-3)*a(n-1) +(n-1)*(n^2-5*n+5)*a(n-2)

%p +2*(n-1)*(n-2)*(n-3)*(a(n-4)-a(n-3)))/(n-2))

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jun 22 2015

%t CoefficientList[Series[E^(x-Sqrt[1-x^2]+1), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jun 22 2015 *)

%K nonn

%O 0,3

%A _Karol A. Penson_, Jun 22 2015