OFFSET
0,4
LINKS
Eric Weisstein's World of Mathematics, Bell Polynomial.
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Emanuele Munarini, Dec 16 2014
STATUS
approved