A363592 Number of partitions of [n] such that in each block the smallest element has the same parity as the largest element.
1, 1, 1, 3, 6, 20, 55, 223, 761, 3595, 14532, 77818, 361605, 2155525, 11274781, 73822175, 428004750, 3046519516, 19348533739, 148493347507, 1023481273549, 8412534272415, 62450994058052, 546699337652602, 4343869829492281, 40308548641909593, 340994681344324137
Offset: 0
Keywords
Examples
a(0) = 1: () the empty partition. a(1) = 1: 1. a(2) = 1: 1|2. a(3) = 3: 123, 13|2, 1|2|3. a(4) = 6: 123|4, 13|24, 13|2|4, 1|234, 1|24|3, 1|2|3|4. a(5) = 20: 12345, 1235|4, 123|4|5, 1245|3, 125|3|4, 1345|2, 135|24, 13|24|5, 135|2|4, 13|2|4|5, 15|234, 1|234|5, 145|2|3, 15|24|3, 1|24|35, 1|24|3|5, 1|2|345, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..150
- Wikipedia, Partition of a set
Programs
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Maple
b:= proc(n, x, y, u, v) option remember; `if`(y+u>n, 0, `if`(n=0, 1, `if`(y=0, 0, b(n-1, v, u, y-1, x+1)*y)+b(n-1, v, u, y, x+1)+ `if`(v=0, 0, b(n-1, v-1, u+1, y, x)*v)+b(n-1, v, u, y, x)*(u+x))) end: a:= n-> b(n, 0$4): seq(a(n), n=0..30);
Formula
a(n) mod 2 = A131719(n+1).