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A363071 Number of partitions of [n] into m blocks that are ordered with increasing least elements and where block j contains n+1-j (m in {0..ceiling(n/2)}, j in {1..m}).

%I A363071 #22 May 17 2023 10:26:48
%S A363071 1,1,1,2,3,6,13,31,80,222,659,2082,6966,24574,91067,353443,1432909,
%T A363071 6054025,26599192,121295345,573065538,2800640187,14137645933,
%U A363071 73619324824,394979697320,2180911872495,12380240599262,72181691321844,431857838950302,2649144684462775
%N A363071 Number of partitions of [n] into m blocks that are ordered with increasing least elements and where block j contains n+1-j (m in {0..ceiling(n/2)}, j in {1..m}).
%H A363071 Alois P. Heinz, <a href="/A363071/b363071.txt">Table of n, a(n) for n = 0..665</a>
%H A363071 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363071 a(n) = Sum_{j=0..ceiling(n/2)} (Stirling2(n-j,j) + Stirling2(n-j,j-1)).
%F A363071 a(n) = A171367(n) + A171367(n-1).
%e A363071 a(0) = 1: (), the empty partition.
%e A363071 a(1) = 1: 1.
%e A363071 a(2) = 1: 12.
%e A363071 a(3) = 2: 123, 13|2.
%e A363071 a(4) = 3: 1234, 124|3, 14|23.
%e A363071 a(5) = 6: 12345, 1235|4, 125|34, 135|24, 15|234, 15|24|3.
%e A363071 a(6) = 13: 123456, 12346|5, 1236|45, 1246|35, 126|345, 126|35|4, 1346|25, 136|245, 136|25|4, 146|235, 16|2345, 16|235|4, 16|25|34.
%e A363071 a(7) = 31: 1234567, 123457|6, 12347|56, 12357|46, 1237|456, 1237|46|5, 12457|36, 1247|356, 1247|36|5, 1257|346, 127|3456, 127|346|5, 127|36|45, 13457|26, 1347|256, 1347|26|5, 1357|246, 137|2456, 137|246|5, 137|26|45, 1457|236, 147|2356, 147|236|5, 157|2346, 17|23456, 17|2346|5, 17|236|45, 147|26|35, 17|246|35, 17|26|345, 17|26|35|4.
%p A363071 b:= proc(n, m) option remember;
%p A363071      `if`(m<n, b(n-1, m)*m+b(n-1, m+1), 1)
%p A363071     end:
%p A363071 a:= n-> b(n, 0):
%p A363071 seq(a(n), n=0..31);
%Y A363071 Cf. A000110, A008277, A048993, A171367, A320964.
%K A363071 nonn
%O A363071 0,4
%A A363071 _Alois P. Heinz_, May 16 2023