Complementary Bell Number

Author

Eric W. Weisstein
January 16, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/ComplementaryBellNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ComplementaryBellNumber.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Sums

Table[Sum[(((-1))^k)StirlingS2[n,k],{k,0,n}],{n,0,10}]

(∑_(k=0)^n)(((-1))^k) (StirlingS2[n, k])==(BellB2[n, -1])

v5

Sum[(((-1))^k)StirlingS2[n,k],{k,0,n}]

v6

Sum[(((-1))^k)StirlingS2[n,k],{k,0,n}]

V7

Sum[(((-1))^k)StirlingS2[n,k],{k,0,n}]

(∑_(k=0)^∞)(((((-1))^k) (k^n))/(k!))==(BellB2[n, -1])

E Table[Sum[(((((-1))^k)(k^n))/(k!)),{k,0,∞}],{n,5}]

v5.2

E Sum[(((((-1))^k)(k^n))/(k!)),{k,0,n}]

v6

E Sum[(((((-1))^k)(k^n))/(k!)),{k,0,∞}]

V7

E Sum[(((((-1))^k)(k^n))/(k!)),{k,0,∞}]

Generating Function

ExponentialGeneratingFunction[BellB[n,-1],n,x]

Table[((BellB[n,-1])/(n!)),{n,0,10}]

Series[(E^(1-(E^x))),{x,0,10}]

SeriesCoefficient[Exp[1-Exp[x]],{x,0,n}]

Closed Forms

(BellB2[n, -1])

FunctionExpand[BellB[n,-1]]

Series[BellB[n,x],{x,0,3}]

BellB[Range[0,10],-1]

Primes

l=BellB[Range[5000],-1];//Timing

pos=Position[l,_?PrimeQ]//Flatten

l[[Take[pos,3]]]