OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = sum((-1)^(n-k)*t(k), k=0..n), where t = A000670 (ordered Bell numbers).
E.g.f.: 1/(2-exp(x))-exp(-x)*log(1/(2-exp(x))). [Typo corrected by Vaclav Kotesovec, Oct 08 2013]
G.f.: 1/(1+x)/Q(0), where Q(k)= 1 - x*(k+1)/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
a(n) ~ n! /(2*(log(2))^(n+1)). - Vaclav Kotesovec, Oct 08 2013
MAPLE
with(combinat):
seq(sum((-1)^(n-k)*sum(factorial(j)*stirling2(k, j), j=0..k), k=0..n), n=0..30); # Muniru A Asiru, Feb 07 2018
MATHEMATICA
t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n - k)t[k], {k, 0, n}], {n, 0, 100}]
(* second program: *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; a[n_] := Sum[(-1)^(n-k) Fubini[k, 1], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 31 2016 *)
PROG
(Maxima)
t(n):=sum(stirling2(n, k)*k!, k, 0, n);
makelist(sum((-1)^(n-k)*t(k), k, 0, n), n, 0, 40);
(Magma)
A000670:=func<n | &+[StirlingSecond(n, i)*Factorial(i): i in [0..n]]>;
[&+[(-1)^(n-k)*A000670(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Oct 03 2012
(PARI) for(n=0, 30, print1(sum(k=0, n, (-1)^(n-k)*sum(j=0, k, j!*stirling(k, j, 2))), ", ")) \\ G. C. Greubel, Feb 07 2018
(GAP) List([0..30], n->Sum([0..n], k->(-1)^(n-k)*Sum([0..k], j-> Factorial(j)*Stirling2(k, j)))); # Muniru A Asiru, Feb 07 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, Oct 02 2012
STATUS
approved