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A363496 Total number of parity changes within the blocks of all partitions of [n].

%I A363496 #15 Jun 12 2023 17:25:36
%S A363496 0,0,1,4,17,74,356,1808,9923,57442,354407,2296028,15704028,112266048,
%T A363496 841442105,6564854864,53413489773,450789496454,3950844987040,
%U A363496 35809477617544,335901221506491,3250110998386534,32453151223493139,333520967584364248,3528754456836294712
%N A363496 Total number of parity changes within the blocks of all partitions of [n].
%H A363496 Alois P. Heinz, <a href="/A363496/b363496.txt">Table of n, a(n) for n = 0..575</a>
%H A363496 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363496 a(n) = Sum_{k=0..max(0,n-1)} k * A363493(n,k).
%e A363496 a(4) = 17 = 6*1 + 4*2 + 1*3: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34, 123|4, 12|34, 14|23, 1|234, 1234.
%p A363496 b:= proc(n, x, y) option remember; `if`(n=0, [1, 0],
%p A363496      `if`(y=0, 0, (p-> p+[0, p[1]])(b(n-1, y-1, x+1)*y))+
%p A363496         b(n-1, y, x)*x + b(n-1, y, x+1))
%p A363496     end:
%p A363496 a:= n-> b(n, 0$2)[2]:
%p A363496 seq(a(n), n=0..24);
%Y A363496 Cf. A000110, A077613, A363493, A363549.
%K A363496 nonn
%O A363496 0,4
%A A363496 _Alois P. Heinz_, Jun 05 2023