A275312 Number of set partitions of [n] with increasing block sizes.

1, 1, 1, 2, 2, 6, 11, 28, 51, 242, 532, 1545, 6188, 16592, 86940, 302909, 967523, 3808673, 23029861, 71772352, 484629531, 1840886853, 9376324526, 37878035106, 204542429832, 1458360522892, 6241489795503, 45783932444672, 211848342780210, 1137580874772724

Offset: 0

Author

Alois P. Heinz, Jul 22 2016

Keywords

• nonn

Examples

			a(4) = 2: 1234, 1|234.
a(5) = 6: 12345, 12|345, 13|245, 14|235, 15|234, 1|2345.
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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..728
• Wikipedia, Partition of a set

Crossrefs

Cf. A007837, A038041, A275309, A275310, A275311, A275313, A286075.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+b(n-i, i+1)*binomial(n-1, i-1)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..35);

Mathematica

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 *)