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A350648 Sum over all partitions of [n] of the number of blocks containing their own index when blocks are ordered with decreasing largest elements.

%I A350648 #17 Jan 11 2022 18:33:21
%S A350648 0,1,1,5,11,48,173,795,3719,19343,106563,628508,3923602,25875858,
%T A350648 179468739,1305268102,9925892324,78728325373,649856661196,
%U A350648 5571421770478,49521735963376,455616186779543,4332419124871058,42520560822961111,430191406640367880
%N A350648 Sum over all partitions of [n] of the number of blocks containing their own index when blocks are ordered with decreasing largest elements.
%H A350648 Alois P. Heinz, <a href="/A350648/b350648.txt">Table of n, a(n) for n = 0..576</a>
%H A350648 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A350648 a(n) = Sum_{k=1..ceiling(n/2)} k * A350647(n,k).
%e A350648 a(3) = 5 = 3*1 + 2*2: 321, 3|21, 3|2|1; 31|2.
%e A350648 a(4) = 11 = 7*1 + 2*2: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2; 431|2, 41|32.
%p A350648 b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p->p+
%p A350648       [0, `if`(j=n, p[1], 0)])(b(n-1, max(j, m))), j=1..m+1))
%p A350648     end:
%p A350648 a:= n-> b(n, 0)[2]:
%p A350648 seq(a(n), n=0..30);
%t A350648 b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[j == n, p[[1]], 0]}][b[n - 1, Max[j, m]]], {j, 1, m + 1}]];
%t A350648 a[n_] := b[n, 0][[2]];
%t A350648 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 11 2022, after _Alois P. Heinz_ *)
%Y A350648 Cf. A350589, A350647.
%K A350648 nonn
%O A350648 0,4
%A A350648 _Alois P. Heinz_, Jan 09 2022