A124325 Number of blocks of size >1 in all partitions of an n-set.

0, 0, 1, 4, 17, 76, 362, 1842, 9991, 57568, 351125, 2259302, 15288000, 108478124, 805037105, 6233693772, 50257390937, 421049519856, 3659097742426, 32931956713294, 306490813820239, 2945638599347760, 29198154161188501

Offset: 0

Author

Emeric Deutsch, Oct 28 2006

Keywords

• nonn

Comments

Sum of the first entries in all blocks of all set partitions of [n-1]. a(4) = 17 because the sum of the first entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+4+3+3+6 = 17. - Alois P. Heinz, Apr 24 2017

Examples

			a(3) = 4 because in the partitions 123, 12|3, 13|2, 1|23, 1|2|3 we have four blocks of size >1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A000110, A123324, A124327, A285595.

Column k=2 of A283424.

Programs

Maple

with(combinat): c:=n->bell(n+1)-bell(n)-n*bell(n-1): seq(c(n),n=0..23);

Mathematica

nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x]-1-x)Exp[Exp[x]-1],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 28 2013 *)

PARI

N = 66;  x = 'x + O('x^N);
egf = (exp(x)-1-x)*exp(exp(x)-1) + 'c0;
gf = serlaplace(egf);
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Mar 29 2013 */

Formula

a(n) = B(n+1)-B(n)-n*B(n-1), where B(q) are the Bell numbers (A000110).

E.g.f.: (exp(z)-1-z)*exp(exp(z)-1).

a(n) = Sum_{k=0..floor(n/2)} k*A124324(n,k).

a(n) = A285595(n-1,1). - Alois P. Heinz, Apr 24 2017

a(n) = Sum_{k=1..n*(n-1)/2} k * A124327(n-1,k) for n>1. - Alois P. Heinz, Dec 05 2023