A363495 Number of partitions of [2n+1] having exactly n parity changes within their blocks.

1, 2, 17, 202, 3899, 98282, 3270604, 134513166, 6744026175, 400657370384, 27819913699591, 2222485356153758, 202085549223540498, 20700107045049813072, 2369116259054858660518, 300712325745715659503258, 42064844140178917094949029, 6448050588990736076081469470

Offset: 0

Author

Alois P. Heinz, Jun 05 2023

Keywords

• nonn

Examples

			a(0) = 1: 1.
a(1) = 2: 12|3, 1|23.
a(2) = 17: 1235|4, 123|4|5, 1245|3, 12|34|5, 125|3|4, 12|3|45, 1345|2, 134|25, 14|235, 14|23|5, 15|234, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..110
• Wikipedia, Partition of a set

Crossrefs

Cf. A363493.

Programs

Maple

b:= proc(n, x, y) option remember; `if`(n=0, 1,
     `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))+
        b(n-1, y, x)*x + b(n-1, y, x+1))
    end:
a:= n-> coeff(b(2*n+1, 0$2),z,n):
seq(a(n), n=0..17);

Formula

a(n) = A363493(2n+1,n).