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A275311 Number of set partitions of [n] with nondecreasing block sizes.

%I A275311 #13 May 01 2017 15:23:12
%S A275311 1,1,2,3,7,12,43,89,363,1096,4349,14575,77166,265648,1369284,6700177,
%T A275311 33526541,162825946,1034556673,5157939218,33054650345,206612195885,
%U A275311 1244742654646,8071979804457,62003987375957,381323590616995,2827411772791596,22061592185044910
%N A275311 Number of set partitions of [n] with nondecreasing block sizes.
%H A275311 Alois P. Heinz, <a href="/A275311/b275311.txt">Table of n, a(n) for n = 0..617</a>
%H A275311 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A275311 a(3) = 3: 123, 1|23, 1|2|3.
%e A275311 a(4) = 7: 1234, 12|34, 13|24, 14|23, 1|234, 1|2|34, 1|2|3|4.
%e A275311 a(5) = 12: 12345, 12|345, 13|245, 14|235, 15|234, 1|2345, 1|23|45, 1|24|35, 1|25|34, 1|2|345, 1|2|3|45, 1|2|3|4|5.
%p A275311 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A275311       add(b(n-j, j)*binomial(n-1, j-1), j=i..n))
%p A275311     end:
%p A275311 a:= n-> b(n, 1):
%p A275311 seq(a(n), n=0..35);
%t A275311 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, j]*Binomial[n-1, j-1], {j, i, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jan 22 2017, translated from Maple *)
%Y A275311 Cf. A007837, A038041, A275309, A275310, A275312, A275313, A286074.
%K A275311 nonn
%O A275311 0,3
%A A275311 _Alois P. Heinz_, Jul 22 2016