%I #11 Jan 05 2024 07:55:09
%S 1,2,8,34,156,815,4946,34706,277768,2500077,25000990,275011176,
%T 3300134476,42901748643,600624481562,9009367224110,144149875586576,
%U 2450547884972761,44109861929510838,838087376660707252,16761747533214146580,351996698197497079951
%N a(n) = n! * (1 + Sum_{k=0..n} binomial(k+2,3) / k!).
%F a(0) = 1; a(n) = n*a(n-1) + binomial(n+2,3).
%F a(n) = n! + A368574(n).
%F E.g.f.: (1 + x * (1+x+x^2/6) * exp(x)) / (1-x).
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x*sum(k=0, 2, binomial(2, k)*x^k/(k+1)!)*exp(x))/(1-x)))
%Y Cf. A033540, A368762, A368764.
%Y Cf. A368574, A368767.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jan 04 2024