%I #35 May 30 2019 16:56:50
%S 1,1,9,121,2161,48081,1279801,39631369,1398961761,55422807841,
%T 2434261023721,117366299630361,6161301265353169,349768597919934961,
%U 21347094823271661081,1393695557886847095721,96910923898115717350081,7149718240571434690591809
%N E.g.f.: exp(x/(1 - 4*x)).
%C For k = 2, 3, 4, ... the difference a(n+k) - a(n) is divisible by k.
%H Ludovic Schwob, <a href="/A321847/b321847.txt">Table of n, a(n) for n = 0..199</a>
%H Norihiro Nakashima, Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%F a(n) = Sum_{k=0..n} 4^(n - k)*(n!/k!)*binomial(n-1, k-1).
%F Recurrence: a(n) = (8*n - 7)*a(n-1) - 16*(n-2)*(n-1)*a(n-2).
%F a(n) ~ n! * exp(sqrt(n) - 1/8) * 2^(2*n - 3/2) / (sqrt(Pi) * n^(3/4)). - _Vaclav Kotesovec_, Nov 21 2018
%p seq(coeff(series(factorial(n)*exp(x/(1-4*x)),x,n+1), x, n), n = 0 .. 17); # _Muniru A Asiru_, Nov 24 2018
%t a[n_] := Sum[4^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (8n - 7)*a[n - 1] - 16(n - 2)(n - 1) *a[n - 2]; Array[a, 20, 0] (* _Amiram Eldar_, Nov 19 2018 *)
%t CoefficientList[Series[Exp[x/(1 - 4*x)], {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* _Stefano Spezia_, Dec 07 2018 *)
%o (PARI) my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-4*x)))) \\ _Michel Marcus_, Nov 25 2018
%o (Maxima) (a[0] : 1, a[1] : 1, a[n] := (8*n - 7)*a[n-1] - 16*(n-2)*(n-1)*a[n-2], makelist(a[n], n, 0, 20)); /* _Franck Maminirina Ramaharo_, Nov 27 2018 */
%Y Cf. A000262, A025168, A321837, A321848, A321849, A321850.
%K nonn
%O 0,3
%A _Ludovic Schwob_, Nov 19 2018