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A362638 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at most i.

%I A362638 #9 May 15 2023 16:28:55
%S A362638 1,1,1,2,4,11,34,117,458,1960,9113,45730,246220,1411820,8584314,
%T A362638 55121249,372445462,2639750490,19570232608,151390999949,1219308754065,
%U A362638 10204045406690,88571532840007,796100505025761,7398510884959638,70995106473861076,702542384597885615
%N A362638 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at most i.
%H A362638 Alois P. Heinz, <a href="/A362638/b362638.txt">Table of n, a(n) for n = 0..578</a>
%H A362638 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A362638 a(0) = 1: ().
%e A362638 a(1) = 1: 1.
%e A362638 a(2) = 1: 1|2.
%e A362638 a(3) = 2: 1|23, 1|2|3.
%e A362638 a(4) = 4: 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
%e A362638 a(5) = 11: 1|23|45, 1|23|4|5, 1|24|35, 1|24|3|5, 1|25|34, 1|2|345, 1|2|34|5, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
%e A362638 a(6) = 34: 1|23|456, 1|23|45|6, 1|23|46|5, 1|23|4|56, 1|23|4|5|6, 1|24|356, 1|24|35|6, 1|24|36|5, 1|24|3|56, 1|24|3|5|6, 1|25|346, 1|25|34|6, 1|26|345, 1|2|345|6, 1|26|34|5, 1|2|346|5, 1|2|34|56, 1|2|34|5|6, 1|25|36|4, 1|25|3|46, 1|25|3|4|6, 1|26|35|4, 1|2|356|4, 1|2|35|46, 1|2|35|4|6, 1|26|3|45, 1|2|36|45, 1|2|3|456, 1|2|3|45|6, 1|26|3|4|5, 1|2|36|4|5, 1|2|3|46|5, 1|2|3|4|56, 1|2|3|4|5|6.
%p A362638 b:= proc(n, t) option remember; `if`(n=0, 1, add(
%p A362638       b(n-j, t+1)*binomial(n-1, j-1), j=1..min(t, n)))
%p A362638     end:
%p A362638 a:= n-> b(n, 1):
%p A362638 seq(a(n), n=0..30);
%Y A362638 Cf. A000110, A362635.
%K A362638 nonn
%O A362638 0,4
%A A362638 _Alois P. Heinz_, Apr 28 2023