A224244 Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique and it contains the element 1.

1, 1, 2, 2, 9, 17, 63, 261, 1088, 4374, 24583, 133861, 740303, 4514824, 29945555, 205127474, 1464586617, 10971233035, 86410874373, 708423380237, 6026435657580, 53117555943951, 485246803230148, 4589013046619689, 44819208415713035, 451184268041122808

Offset: 1

Author

Geoffrey Critzer, Apr 01 2013

Keywords

• nonn

Examples

			a(5) = 9 because we have: {{1,2,3,4,5}}, {{1},{2,3,4,5}}, {{1,2},{3,4,5}}, {{1,3},{2,4,5}}, {{1,5},{2,3,4}}, {{1,4},{2,3,5}}, {{1},{2,3},{4,5}}, {{1},{2,5},{3,4}}, {{1},{2,4},{3,5}}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..578
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 139.

Crossrefs

Cf. A224219.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, add(
      binomial(n-1, i-1)*b(n-i, `if`(t=1, i+1, t)), i=t..n))
    end:
a:= n-> `if`(n=0, 0, b(n, 1)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jul 07 2016

Mathematica

nn=20;Drop[Range[0,nn]!CoefficientList[Series[Sum[Integrate[x^(k-1)/(k-1)! Exp[Exp[x]-Sum[x^i/i!,{i,0,k}]],x],{k,1,nn}],{x,0,nn}],x],1]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n==0, 1, Sum[Binomial[n-1, i-1]*b[n-i, If[t==1, i + 1, t]], {i, t, n}]]; a[n_] := If[n==0, 0, b[n, 1]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

Formula

E.g.f.: Sum_{k>=1} Integral of x^(k-1)/(k-1)! * exp(exp(x) - Sum_{i=0..k} x^i/i!) dx.