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A362637 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i (except possibly the last) has size at least i.

%I A362637 #13 May 15 2023 16:37:59
%S A362637 1,1,2,4,10,30,96,323,1184,4784,20708,93073,431004,2080610,10615276,
%T A362637 57291063,322921896,1871715144,11065738360,66843918825,415837464280,
%U A362637 2684434034706,18010208402784,124877499979859,886741484322660,6399683149311272,46802092819866340
%N A362637 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i (except possibly the last) has size at least i.
%H A362637 Alois P. Heinz, <a href="/A362637/b362637.txt">Table of n, a(n) for n = 0..695</a>
%H A362637 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A362637 a(0) = 1: (), the empty partition.
%e A362637 a(1) = 1: 1.
%e A362637 a(2) = 2: 12, 1|2.
%e A362637 a(3) = 4: 123, 12|3, 13|2, 1|23.
%e A362637 a(4) = 10: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 1|23|4, 1|24|3.
%e A362637 a(5) = 30: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 14|25|3, 15|24|3, 1|245|3, 1|24|35, 1|25|34.
%p A362637 b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<=t, 1,
%p A362637       add(b(n-j, t+1)*binomial(n-1, j-1), j=t..n)))
%p A362637     end:
%p A362637 a:= n-> b(n, 1):
%p A362637 seq(a(n), n=0..30);
%Y A362637 Cf. A000110, A362549, A362635, A362639.
%K A362637 nonn
%O A362637 0,3
%A A362637 _Alois P. Heinz_, Apr 28 2023