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A350650 Number of partitions of [n] having exactly one block containing its own index when blocks are ordered with decreasing largest elements.

%I A350650 #17 May 16 2022 04:58:46
%S A350650 0,1,1,3,7,25,91,390,1797,9069,49106,284537,1751554,11406588,78254594,
%T A350650 563642925,4249337018,33443545866,274130245342,2335311549498,
%U A350650 20637538548167,188867393030394,1787189672368355,17461684290203403,175930808241047092,1825666076751872506
%N A350650 Number of partitions of [n] having exactly one block containing its own index when blocks are ordered with decreasing largest elements.
%H A350650 Alois P. Heinz, <a href="/A350650/b350650.txt">Table of n, a(n) for n = 0..576</a>
%H A350650 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A350650 a(4) = 7: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2.
%p A350650 b:= proc(n, m) option remember; series(`if`(n=0, 1, add(
%p A350650       `if`(j=n, x, 1)*b(n-1, max(m, j)), j=1..m+1)), x, 2)
%p A350650     end:
%p A350650 a:= n-> coeff(b(n, 0), x, 1):
%p A350650 seq(a(n), n=0..25);
%t A350650 b[n_, m_] := b[n, m] = Series[If[n == 0, 1, Sum[
%t A350650     If[j == n, x, 1]*b[n-1, Max[m, j]], {j, 1, m+1}]], {x, 0, 2}];
%t A350650 a[n_] := Coefficient[b[n, 0], x, 1];
%t A350650 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)
%Y A350650 Column k=1 of A350647, A350684.
%K A350650 nonn
%O A350650 0,4
%A A350650 _Alois P. Heinz_, Jan 09 2022