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

0,5

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).

Links

Table of n, a(n) for 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

Maple

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

Crossrefs

Cf. A000110, A005001.

Sequence in context: A289779 A117423 A351053 * A338584 A084778 A287807

Adjacent sequences: A171856 A171857 A171858 * A171860 A171861 A171862

Keyword

nonn

Author

Emeric Deutsch, May 01 2010

Status

approved