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A124325 Number of blocks of size >1 in all partitions of an n-set.

%I A124325 #19 Dec 05 2023 19:25:47
%S A124325 0,0,1,4,17,76,362,1842,9991,57568,351125,2259302,15288000,108478124,
%T A124325 805037105,6233693772,50257390937,421049519856,3659097742426,
%U A124325 32931956713294,306490813820239,2945638599347760,29198154161188501
%N A124325 Number of blocks of size >1 in all partitions of an n-set.
%C A124325 Sum of the first entries in all blocks of all set partitions of [n-1]. a(4) = 17 because the sum of the first entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+4+3+3+6 = 17. - _Alois P. Heinz_, Apr 24 2017
%H A124325 Alois P. Heinz, <a href="/A124325/b124325.txt">Table of n, a(n) for n = 0..400</a>
%F A124325 a(n) = B(n+1)-B(n)-n*B(n-1), where B(q) are the Bell numbers (A000110).
%F A124325 E.g.f.: (exp(z)-1-z)*exp(exp(z)-1).
%F A124325 a(n) = Sum_{k=0..floor(n/2)} k*A124324(n,k).
%F A124325 a(n) = A285595(n-1,1). - _Alois P. Heinz_, Apr 24 2017
%F A124325 a(n) = Sum_{k=1..n*(n-1)/2} k * A124327(n-1,k) for n>1. - _Alois P. Heinz_, Dec 05 2023
%e A124325 a(3) = 4 because in the partitions 123, 12|3, 13|2, 1|23, 1|2|3 we have four blocks of size >1.
%p A124325 with(combinat): c:=n->bell(n+1)-bell(n)-n*bell(n-1): seq(c(n),n=0..23);
%t A124325 nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x]-1-x)Exp[Exp[x]-1],{x,0,nn}],x]  (* _Geoffrey Critzer_, Mar 28 2013 *)
%o A124325 (PARI)
%o A124325 N = 66;  x = 'x + O('x^N);
%o A124325 egf = (exp(x)-1-x)*exp(exp(x)-1) + 'c0;
%o A124325 gf = serlaplace(egf);
%o A124325 v = Vec(gf);  v[1]-='c0;  v
%o A124325 /* _Joerg Arndt_, Mar 29 2013 */
%Y A124325 Cf. A000110, A123324, A124327, A285595.
%Y A124325 Column k=2 of A283424.
%K A124325 nonn
%O A124325 0,4
%A A124325 _Emeric Deutsch_, Oct 28 2006