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A262321 Number of ways to select a subset s containing n from {1,...,n} and then partition s into blocks of equal size.

%I A262321 #15 Feb 16 2019 19:48:21
%S A262321 1,1,3,7,18,43,118,337,1025,3479,13056,48817,199477,898135,4051128,
%T A262321 18652459,93872040,492132207,2658676056,14841915049,84757413959,
%U A262321 517609038551,3384739112196,21742333893177,141230605251082,1001795869162783,7387581072984938
%N A262321 Number of ways to select a subset s containing n from {1,...,n} and then partition s into blocks of equal size.
%C A262321 a(0) = 1 by convention.
%H A262321 Alois P. Heinz, <a href="/A262321/b262321.txt">Table of n, a(n) for n = 0..616</a>
%F A262321 E.g.f.: A(x) - Integral_{x} A(x) dx, with A(x) = e.g.f. of A262320.
%e A262321 a(0) = 1: {}.
%e A262321 a(1) = 1: 1.
%e A262321 a(2) = 3: 2, 12, 1|2.
%e A262321 a(3) = 7: 3, 13, 1|3, 23, 2|3, 123, 1|2|3.
%e A262321 a(4) = 18: 4, 14, 1|4, 24, 2|4, 34, 3|4, 124, 1|2|4, 134, 1|3|4, 234, 2|3|4, 1234, 12|34, 13|24, 14|23, 1|2|3|4.
%p A262321 b:= proc(n) option remember; n!*`if`(n=0, 1,
%p A262321        add(1/(d!*(n/d)!^d), d=numtheory[divisors](n)))
%p A262321     end:
%p A262321 a:= n-> add(b(k)*binomial(n-1, k-1), k=0..n):
%p A262321 seq(a(n), n=0..30);
%t A262321 b[n_] := b[n] = n!*If[n == 0, 1, DivisorSum[n, 1/(#!*(n/#)!^#)&]];
%t A262321 a[n_] :=  Sum[b[k]*Binomial[n-1, k-1], {k, 0, n}];
%t A262321 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 28 2017, translated from Maple *)
%Y A262321 First differences of A262320.
%K A262321 nonn
%O A262321 0,3
%A A262321 _Alois P. Heinz_, Sep 18 2015