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A224219 Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique.

%I A224219 #23 May 13 2024 15:12:52
%S A224219 1,1,4,5,31,82,344,1661,7942,38721,228680,1377026,8529756,56756260,
%T A224219 402300799,2960135917,22692746719,181667760724,1516381486766,
%U A224219 13135566948285,117868982320877,1093961278908818,10492653292100919,103880022098900234,1059925027073166856
%N A224219 Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique.
%C A224219 In other words, if the smallest block in a partition has size k then there are no other blocks in the partition with size k.
%H A224219 Alois P. Heinz, <a href="/A224219/b224219.txt">Table of n, a(n) for n = 1..576</a>
%F A224219 E.g.f.: Sum_{k>=1} x^k/k! * exp(exp(x) - Sum_{i=0..k} x^i/i!).
%e A224219 a(4) = 5 because we have: {{1,2,3,4}}, {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}.
%p A224219 with(combinat):
%p A224219 b:= proc(n, i) option remember;
%p A224219       `if`(i<1, 0, `if`(n=i, 1, 0)+add(b(n-i*j, i-1)*
%p A224219        multinomial(n, n-i*j, i$j)/j!, j=0..(n-1)/i))
%p A224219     end:
%p A224219 a:= n-> b(n$2):
%p A224219 seq(a(n), n=1..25);  # _Alois P. Heinz_, Jul 07 2016
%t A224219 nn=25;Drop[Range[0,nn]!CoefficientList[Series[Sum[x^k/k!Exp[Exp[x]-Sum[x^i/i!,{i,0,k}]],{k,1,nn}],{x,0,nn}],x],1]
%t A224219 (* Second program: *)
%t A224219 multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[i<1, 0, If[n==i, 1, 0] + Sum[b[n-i*j, i-1]*multinomial[n, Prepend[Array[i&, j], n-i*j]]/j!, {j, 0, (n-1)/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 25}] (* _Jean-François Alcover_, Feb 03 2017, after _Alois P. Heinz_ *)
%Y A224219 Cf. A224244, A372721, A372802.
%Y A224219 Column k=1 of A372762.
%K A224219 nonn
%O A224219 1,3
%A A224219 _Geoffrey Critzer_, Apr 01 2013