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%I #14 May 01 2017 15:44:12
%S 1,1,1,2,2,6,11,28,51,242,532,1545,6188,16592,86940,302909,967523,
%T 3808673,23029861,71772352,484629531,1840886853,9376324526,
%U 37878035106,204542429832,1458360522892,6241489795503,45783932444672,211848342780210,1137580874772724
%N Number of set partitions of [n] with increasing block sizes.
%H Alois P. Heinz, <a href="/A275312/b275312.txt">Table of n, a(n) for n = 0..728</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e a(4) = 2: 1234, 1|234.
%e a(5) = 6: 12345, 12|345, 13|245, 14|235, 15|234, 1|2345.
%e a(6) = 11: 123456, 12|3456, 13|2456, 14|2356, 15|2346, 16|2345, 1|23456, 1|23|456, 1|24|356, 1|25|346, 1|26|345.
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i>n, 0, b(n, i+1)+b(n-i, i+1)*binomial(n-1, i-1)))
%p end:
%p a:= n-> b(n, 1):
%p seq(a(n), n=0..35);
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i+1] + b[n-i, i+1] * Binomial[n-1, i-1]]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jan 22 2017, translated from Maple *)
%Y Cf. A007837, A038041, A275309, A275310, A275311, A275313, A286075.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Jul 22 2016
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