OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..200
V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013
FORMULA
a(n) = Sum_{k>=0} binomial(n-k+1,k)*Stirling2(n,k)*k!.
Limit n->infinity (a(n)/n!)^(1/n) = (3*r^2-3*r+1)/(1-2*r) = 1.53445630931668421506236..., where r = 0.410751485627... is the root of the equation (1-2*r)^2 + r*(1-3*r+3*r^2)*LambertW(-exp(-1/r)/r) = 0. - Vaclav Kotesovec, Dec 08 2012
EXAMPLE
a(3) = 9 because we have: (1,1,3), (1,3,1), (1,3,3), (3,1,1), (3,1,3), (3,3,1), (1,1,1), (2,2,2), (3,3,3).
MAPLE
with (combinat):
a:= n-> add(stirling2(n, k)*k! *binomial(n-k+1, k), k=0..ceil(n/2)):
seq (a(n), n=0..30); # Alois P. Heinz, Dec 06 2012
MATHEMATICA
Table[Sum[Binomial[n-k+1, k]StirlingS2[n, k]k!, {k, 0, n}], {n, 0, 20}]
(* Program for numerical value of the limit (a(n)/n!)^(1/n) *) (3*r^2-3*r+1)/(1-2*r)/.FindRoot[(1-2*r)^2+r*(1-3*r+3*r^2)*LambertW[-E^(-1/r)/r]==0, {r, 1/2}, WorkingPrecision->100] (* Vaclav Kotesovec, Dec 08 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Dec 06 2012
STATUS
approved