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A367850 Total sum of the block maxima minus the block minima over all partitions of [n].

%I A367850 #48 Feb 02 2025 10:57:25
%S A367850 0,0,1,6,33,182,1034,6122,37927,246030,1669941,11844324,87644672,
%T A367850 675494180,5413500801,45040155758,388441330457,3467619369538,
%U A367850 31998729152474,304846692965822,2994781617653439,30304301968015582,315536869771786501,3377398077726963112
%N A367850 Total sum of the block maxima minus the block minima over all partitions of [n].
%H A367850 Alois P. Heinz, <a href="/A367850/b367850.txt">Table of n, a(n) for n = 0..574</a>
%H A367850 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A367850 E.g.f.: (z-2)*exp(2*z+exp(z)-1)+(2*z+1)*exp(z+exp(z)-1)+exp(exp(z)-1).
%F A367850 a(n) = A278677(n-1) - A124325(n+1) for n>=1.
%F A367850 a(n) = Bell(n+1)+(n+1)*Bell(n)-Bell(n+2)+Sum_{k=0..n} Stirling2(n+1,k)*(n+1-k).
%F A367850 a(n) = Sum_{k=0..A002620(n)} k * A368338(n,k).
%F A367850 a(n) mod 2 = A120325(n).
%e A367850 a(3) = 6 = 2 + 1 + 2 + 1 + 0: 123, 12|3, 13|2, 1|23, 1|2|3.
%p A367850 b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], (p->
%p A367850       p+[0, p[1]*(n-t)])(b(n-1, m+1, t+1))+m*b(n-1, m, t+1))
%p A367850     end:
%p A367850 a:= n-> b(n, 0, 1)[2]:
%p A367850 seq(a(n), n=0..23);
%p A367850 # second Maple program:
%p A367850 egf:= (z-2)*exp(2*z+exp(z)-1)+(2*z+1)*exp(z+exp(z)-1)+exp(exp(z)-1):
%p A367850 a:= n-> n!*coeff(series(egf, z, n+1), z, n):
%p A367850 seq(a(n), n=0..23);
%Y A367850 Cf. A000110, A002538 (the same for permutations), A002620, A120325, A124325, A278677, A368338.
%K A367850 nonn
%O A367850 0,4
%A A367850 _Alois P. Heinz_, Dec 15 2023