A275309 - OEIS

%I #14 May 01 2017 15:14:30

%S 1,1,1,3,4,11,36,82,239,821,3742,10328,42934,156070,729249,4025361,

%T 15032099,68746675,334541624,1645575386,9104991312,65010298257,

%U 282768687257,1616844660914,8660050947383,53262316928024,309119883729116,2185141720645817

%N Number of set partitions of [n] with decreasing block sizes.

%H Alois P. Heinz, <a href="/A275309/b275309.txt">Table of n, a(n) for n = 0..717</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e a(3) = 3: 123, 12|3, 13|2.

%e a(4) = 4: 1234, 123|4, 124|3, 134|2.

%e a(5) = 11: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 1345|2, 134|25, 135|24, 145|23.

%p b:= proc(n, i) option remember;

%p       `if`(n>i*(i+1)/2, 0, `if`(n=0, 1, b(n, i-1)+

%p       `if`(i>n, 0, b(n-i, i-1)*binomial(n-1, i-1))))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..35);

%t b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, b[n, i - 1] +  If[i > n, 0, b[n - i, i - 1]*Binomial[n - 1, i - 1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jan 21 2017, translated from Maple *)

%Y Cf. A007837, A038041, A275310, A275311, A275312, A275313, A286073.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jul 22 2016