A252284 - OEIS

%I #28 Jan 31 2024 08:05:49

%S 1,-1,-1,3,9,-21,-129,111,2577,2871,-57249,-232101,1175769,11951523,

%T -6313761,-542318841,-1778088159,20647593711,187318128447,

%U -386536525389,-13793029404759,-41926398389541,783578974052799,7433562140085663,-22263437361406671,-767083139039850201

%N Exponential generating function exp(-x-x^2-x^3/3).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

%F a(n) = Sum_{k=0..n} ((n!/k!)*(-1)^k * Sum_{i=0..k} C(k,i)*C(k-i,n-2*i-k)/3^i).

%F E.g.f.: exp(-x-x^2-x^3/3).

%F Recurrence: a(n+3)+a(n+2)+2*(n+2)*a(n+1)+(n+2)*(n+1)*a(n)=0.

%F a(n) = Sum_{k=0..n} 3^k * Stirling1(n,k) * Bell_k(-1/3), where Bell_n(x) is n-th Bell polynomial. - _Seiichi Manyama_, Jan 31 2024

%p S:= series(exp(-x-x^2-x^3/3),x,101):

%p seq(coeff(S,x,j)*j!,j=0..100); # _Robert Israel_, Dec 16 2014

%t a[n_] := Sum[(n!/k!)(-1)^k Sum[Binomial[k,i]Binomial[k-i,n-2i-k]/3^i,{i,0,k}],{k,0,n}]; Table[a[n],{n,0,20}]

%t With[{nn=30},CoefficientList[Series[Exp[-x-x^2-x^3/3],{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jan 01 2021 *)

%o (Maxima) a(n) := sum((n!/k!)*(-1)^k*sum(binomial(k,i)*binomial(k-i,n-2*i-k)/3^i,i,0,k),k,0,n);

%o makelist(a(n),n,0,20);

%o (PARI) default(seriesprecision, 40); Vec(serlaplace( exp(-x-x^2-x^3/3))) \\ _Michel Marcus_, Dec 17 2014

%Y Cf. A001464, A249015.

%K sign

%O 0,4

%A _Emanuele Munarini_, Dec 16 2014