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A372650 Total sum over all partitions of [n] of the number of minimal blocks.

%I A372650 #18 May 12 2024 14:20:23
%S A372650 0,1,3,7,27,86,393,1688,8291,44143,248428,1480073,9440049,63265606,
%T A372650 444309232,3273807272,25227429123,202458174614,1689474026499,
%U A372650 14636685675142,131413462612012,1220636654904548,11712836883408675,115956109213769404,1182944504931376337
%N A372650 Total sum over all partitions of [n] of the number of minimal blocks.
%H A372650 Alois P. Heinz, <a href="/A372650/b372650.txt">Table of n, a(n) for n = 0..576</a>
%H A372650 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A372650 a(n) = Sum_{k=0..n} k * A372762(n,k).
%e A372650 a(4) = 27 = 1+1+1+2+2+1+2+2+2+1+2+2+2+2+4: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
%p A372650 b:= proc(n, m, t) option remember; `if`(n=0, t,
%p A372650       add(binomial(n-1, j-1)*b(n-j, min(j, m),
%p A372650      `if`(j<m, 1, `if`(j=m, t+1, t))), j=1..n))
%p A372650     end:
%p A372650 a:= n-> b(n$2, 0):
%p A372650 seq(a(n), n=0..24);
%t A372650 b[n_, m_, t_] := b[n, m, t] = If[n == 0, t,
%t A372650    Sum[Binomial[n - 1, j - 1]*b[n - j, Min[j, m],
%t A372650    If[j < m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
%t A372650 a[n_] := b[n, n, 0];
%t A372650 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, May 11 2024, after _Alois P. Heinz_ *)
%Y A372650 Cf. A000110, A097147, A372649, A372762.
%K A372650 nonn
%O A372650 0,3
%A A372650 _Alois P. Heinz_, May 08 2024