A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).

0, 0, 0, 1, 8, 47, 258, 1426, 8154, 48715, 305012, 2001719, 13754692, 98801976, 740584196, 5782218745, 46942426080, 395607218279, 3455493024350, 31236784338746, 291836182128670, 2814329123555051, 27980637362452980

Offset: 0

Author

Emeric Deutsch, May 07 2010

Keywords

• nonn

Comments

Number of blocks not consisting of consecutive integers in all partitions of the set {1,2,...,n} (a singleton is considered a block of consecutive integers). Example: a(3)=1 because in 1-2-3, 1-23, 12-3, 13-2, and 123 only the block 13 does not consist of consecutive integers.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000110, A005493, A177254, A177255, A177256.

Programs

Magma

A177257:= func< n | n eq 0 select 0 else (&+[(Binomial(n,j)-(j+1))*Bell(j): j in [0..n-1]]) >;
[A177257(n): n in [0..30]]; // G. C. Greubel, May 12 2024

Maple

with(combinat): a:= proc(n) add((binomial(n, j)-j-1)*bell(j), j = 0 .. n-1) end proc: seq(a(n), n = 0 .. 22);

Mathematica

Table[Sum[(Binomial[n,j]-j-1)BellB[j],{j,0,n-1}],{n,0,30}] (* Harvey P. Dale, Oct 15 2015 *)

SageMath

def A177257(n): return sum((binomial(n,j) -(j+1))*bell_number(j) for j in range(n))
[A177257(n) for n in range(31)] # G. C. Greubel, May 12 2024

Formula

a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*Bell(j), where Bell(n) = A000110(n) are the Bell numbers.

a(n) = Sum_{k=0..floor(n/2)} k*A177256(n,k).

a(n) = A005493(n-1) - A177255(n).