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

%I A363495 #14 Jun 06 2023 17:05:33
%S A363495 1,2,17,202,3899,98282,3270604,134513166,6744026175,400657370384,
%T A363495 27819913699591,2222485356153758,202085549223540498,
%U A363495 20700107045049813072,2369116259054858660518,300712325745715659503258,42064844140178917094949029,6448050588990736076081469470
%N A363495 Number of partitions of [2n+1] having exactly n parity changes within their blocks.
%H A363495 Alois P. Heinz, <a href="/A363495/b363495.txt">Table of n, a(n) for n = 0..110</a>
%H A363495 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363495 a(n) = A363493(2n+1,n).
%e A363495 a(0) = 1: 1.
%e A363495 a(1) = 2: 12|3, 1|23.
%e A363495 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.
%p A363495 b:= proc(n, x, y) option remember; `if`(n=0, 1,
%p A363495      `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))+
%p A363495         b(n-1, y, x)*x + b(n-1, y, x+1))
%p A363495     end:
%p A363495 a:= n-> coeff(b(2*n+1, 0$2),z,n):
%p A363495 seq(a(n), n=0..17);
%Y A363495 Cf. A363493.
%K A363495 nonn
%O A363495 0,2
%A A363495 _Alois P. Heinz_, Jun 05 2023