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

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..578

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 139.

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.

Example

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}}.

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 *)

Crossrefs

Cf. A224219.

Sequence in context: A309705 A290604 A039796 * A007024 A019223 A192302

Adjacent sequences: A224241 A224242 A224243 * A224245 A224246 A224247

Keyword

nonn

Author

Geoffrey Critzer, Apr 01 2013

Status

approved