A350650 - OEIS

%I #17 May 16 2022 04:58:46

%S 0,1,1,3,7,25,91,390,1797,9069,49106,284537,1751554,11406588,78254594,

%T 563642925,4249337018,33443545866,274130245342,2335311549498,

%U 20637538548167,188867393030394,1787189672368355,17461684290203403,175930808241047092,1825666076751872506

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

%H Alois P. Heinz, <a href="/A350650/b350650.txt">Table of n, a(n) for n = 0..576</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e a(4) = 7: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2.

%p b:= proc(n, m) option remember; series(`if`(n=0, 1, add(

%p       `if`(j=n, x, 1)*b(n-1, max(m, j)), j=1..m+1)), x, 2)

%p     end:

%p a:= n-> coeff(b(n, 0), x, 1):

%p seq(a(n), n=0..25);

%t b[n_, m_] := b[n, m] = Series[If[n == 0, 1, Sum[

%t     If[j == n, x, 1]*b[n-1, Max[m, j]], {j, 1, m+1}]], {x, 0, 2}];

%t a[n_] := Coefficient[b[n, 0], x, 1];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)

%Y Column k=1 of A350647, A350684.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jan 09 2022