A171859 a(n) = Bell(n) - Sum_{j=0..n-1} Bell(j), where the Bell numbers are given in A000110.

1, 0, 0, 1, 6, 28, 127, 598, 2984, 15851, 89532, 536152, 3392609, 22609852, 158220300, 1159380201, 8873605258, 70778190768, 587125257319, 5055713850058, 45114387675316, 416535887361323, 3973511993495144, 39112086371684844

Offset: 0

Author

Emeric Deutsch, May 01 2010

Keywords

• nonn

Comments

Number of partitions of the set {1,2,...,n} in which n is neither a singleton nor is in a block of consecutive integers. Example: a(4)=6 because we have 14-23, 13-24, 134-2, 124-3, 1-24-3, and 14-2-3. Note that if from the other partitions of {1,2,3,4}, namely 1234, 1-234, 12-34, 1-2-34, 123-4, 1-23-4, 12-3-4, 13-2-4, 1-2-3-4, we delete the blocks containing 4, then we are left with empty, 1, 12, 1-2, 123, 1-23, 12-3, 13-2, 1-2-3, i.e., all the partitions of the sets: empty, {1}, {1,2}, and {1,2,3}.

a(n) = A000110(n) - A005001(n).

Crossrefs

Cf. A000110, A005001.

Programs

Maple

with(combinat): seq(bell(n)-add(bell(j), j = 0 .. n-1), n = 0 .. 23);

Formula

G.f.: G(0)*(1-x-x^2)/(1-x^2) + x/(1-x^2) where G(k) = 1 - x*(1-k*x)/(1 - x - x^2 - (1-2*x-x^2+2*x^3+x^4)/(1 - x - x^2 + (1-k*x)*(k*x+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 10 2013