%I #20 Oct 20 2024 04:08:12
%S 1,0,2,3,-4,90,-126,-840,21104,-137592,-88920,15741000,-197234808,
%T 535289040,25582565904,-522317151720,3223601137920,75590725210560,
%U -2388641226278976,23718732310200960,361277667059425920,-17515819241263405440,246424647059545933440
%N Expansion of e.g.f. (1 + x + x^2)^x.
%H Robert Israel, <a href="/A191422/b191422.txt">Table of n, a(n) for n = 0..448</a>
%F E.g.f.: (1 + x + x^2)^x.
%F a(n) = n!*Sum_{m=1..n} Sum_{k=0..n-2*m} Stirling1(m+k, m)*binomial(m+k, n-2*m-k)/(m+k)! for n > 0, a(0)=1.
%p S:= series((1+x+x^2)^x,x,41):
%p seq(coeff(S,x,k)*k!,k=0..40); # _Robert Israel_, Apr 28 2021
%o (Maxima)
%o a(n):=if n=0 then 1 else (sum(sum((stirling1(m+k,m)*binomial(m+k,n-2*m-k))/(m+k)!,k,0,n-2*m),m,1,n))*n!;
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace((1+x+x^2)^x)) \\ _Michel Marcus_, Apr 28 2021
%K sign
%O 0,3
%A _Vladimir Kruchinin_, Jun 02 2011