A252284 Exponential generating function exp(-x-x^2-x^3/3).

1, -1, -1, 3, 9, -21, -129, 111, 2577, 2871, -57249, -232101, 1175769, 11951523, -6313761, -542318841, -1778088159, 20647593711, 187318128447, -386536525389, -13793029404759, -41926398389541, 783578974052799, 7433562140085663, -22263437361406671, -767083139039850201

Offset: 0

Author

Emanuele Munarini, Dec 16 2014

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Cf. A001464, A249015.

Programs

Maple

S:= series(exp(-x-x^2-x^3/3),x,101):
seq(coeff(S,x,j)*j!,j=0..100); # Robert Israel, Dec 16 2014

Mathematica

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}]
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 *)

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

PARI

default(seriesprecision, 40); Vec(serlaplace( exp(-x-x^2-x^3/3))) \\ Michel Marcus, Dec 17 2014

Formula

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).

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

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

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