A224244 - OEIS

%I #22 Nov 02 2024 04:06:53

%S 1,1,2,2,9,17,63,261,1088,4374,24583,133861,740303,4514824,29945555,

%T 205127474,1464586617,10971233035,86410874373,708423380237,

%U 6026435657580,53117555943951,485246803230148,4589013046619689,44819208415713035,451184268041122808

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

%H Alois P. Heinz, <a href="/A224244/b224244.txt">Table of n, a(n) for n = 1..578</a>

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; page 139.

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

%e 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}}.

%p b:= proc(n, t) option remember; `if`(n=0, 1, add(

%p       binomial(n-1, i-1)*b(n-i, `if`(t=1, i+1, t)), i=t..n))

%p     end:

%p a:= n-> `if`(n=0, 0, b(n, 1)):

%p seq(a(n), n=1..30);  # _Alois P. Heinz_, Jul 07 2016

%t 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]

%t (* Second program: *)

%t 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_ *)

%Y Cf. A224219.

%K nonn

%O 1,3

%A _Geoffrey Critzer_, Apr 01 2013