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A275310 Number of set partitions of [n] with nonincreasing block sizes.

%I A275310 #13 May 01 2017 14:43:16
%S A275310 1,1,2,4,11,30,102,346,1353,5444,24170,110082,546075,2777828,15099359,
%T A275310 84491723,499665713,3035284304,19375261490,126821116410,866293979945,
%U A275310 6072753348997,44193947169228,329387416656794,2542173092336648,20069525888319293
%N A275310 Number of set partitions of [n] with nonincreasing block sizes.
%H A275310 Alois P. Heinz, <a href="/A275310/b275310.txt">Table of n, a(n) for n = 0..612</a>
%H A275310 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A275310 a(3) = 4: 123, 12|3, 13|2, 1|2|3.
%e A275310 a(4) = 11: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 14|2|3, 1|2|3|4.
%p A275310 b:= proc(n, i) option remember; `if`(n=0, 1, add(
%p A275310       b(n-j, j)*binomial(n-1, j-1), j=1..min(n, i)))
%p A275310     end:
%p A275310 a:= n-> b(n$2):
%p A275310 seq(a(n), n=0..35);
%t A275310 b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j]*Binomial[n-1, j-1], {j, 1, Min[n, i]}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Feb 02 2017, translated from Maple *)
%Y A275310 Cf. A007837, A038041, A275309, A275311, A275312, A275313, A286071.
%K A275310 nonn
%O A275310 0,3
%A A275310 _Alois P. Heinz_, Jul 22 2016