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A362635 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at least i.

%I A362635 #15 May 15 2023 16:13:06
%S A362635 1,1,1,2,5,12,31,97,351,1318,4963,19391,82531,386704,1926907,9811733,
%T A362635 50252175,261462430,1415025895,8118274255,49355434511,312266428040,
%U A362635 2012834117143,13055850467371,85215848844559,565353777291346,3866868949795579,27548261709035105
%N A362635 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at least i.
%H A362635 Alois P. Heinz, <a href="/A362635/b362635.txt">Table of n, a(n) for n = 0..700</a>
%H A362635 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A362635 a(0) = 1: (), the empty partition.
%e A362635 a(1) = 1: 1.
%e A362635 a(2) = 1: 12.
%e A362635 a(3) = 2: 123, 1|23.
%e A362635 a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
%e A362635 a(5) = 12: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
%e A362635 a(6) = 31: 123456, 1234|56, 1235|46, 1236|45, 123|456, 1245|36, 1246|35, 124|356, 1256|34, 125|346, 126|345, 12|3456, 1345|26, 1346|25, 134|256, 1356|24, 135|246, 136|245, 13|2456, 1456|23, 145|236, 146|235, 14|2356, 156|234, 15|2346, 16|2345, 1|23456, 1|23|456, 1|24|356, 1|25|346, 1|26|345.
%p A362635 b:= proc(n, t) option remember; `if`(n=0, 1, add(
%p A362635       b(n-j, t+1)*binomial(n-1, j-1), j=t..n))
%p A362635     end:
%p A362635 a:= n-> b(n, 1):
%p A362635 seq(a(n), n=0..30);
%Y A362635 Cf. A000110, A362549, A362637, A362638, A362893.
%K A362635 nonn
%O A362635 0,4
%A A362635 _Alois P. Heinz_, Apr 28 2023