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A363588 Number of partitions of [n] having exactly two parity changes within their blocks.

%I A363588 #10 Jun 10 2023 17:10:42
%S A363588 0,0,0,1,4,17,68,292,1252,5670,26114,126073,621914,3206277,16888898,
%T A363588 92771126,519907322,3032369590,18012896770,111155162265,697399200274,
%U A363588 4537750415991,29972920817228,204993708306706,1421278374189924,10188372221843166,73948157842293620
%N A363588 Number of partitions of [n] having exactly two parity changes within their blocks.
%H A363588 Alois P. Heinz, <a href="/A363588/b363588.txt">Table of n, a(n) for n = 0..500</a>
%H A363588 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A363588 a(4) = 4: 123|4, 12|34, 14|23, 1|234.
%p A363588 b:= proc(n, x, y) option remember; convert(series(
%p A363588      `if`(n=0, 1, `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))
%p A363588        +b(n-1, y, x)*x + b(n-1, y, x+1)), z, 3), polynom)
%p A363588     end:
%p A363588 a:= n-> coeff(b(n, 0$2), z, 2):
%p A363588 seq(a(n), n=0..27);
%Y A363588 Column k=2 of A363493.
%Y A363588 Cf. A000110.
%K A363588 nonn
%O A363588 0,5
%A A363588 _Alois P. Heinz_, Jun 10 2023