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Number of partitions of [n] having exactly one parity change within their blocks.
2

%I #10 Jun 06 2023 20:00:20

%S 0,0,1,2,6,18,61,210,778,3008,12219,52268,231726,1083012,5202199,

%T 26307710,135972580,738339310,4081523615,23649300862,139096468520,

%U 855529383396,5329630673249,34643027568520,227682351175868,1558106351450416,10766192988109009

%N Number of partitions of [n] having exactly one parity change within their blocks.

%H Alois P. Heinz, <a href="/A363511/b363511.txt">Table of n, a(n) for n = 0..500</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e a(4) = 6: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34.

%p b:= proc(n, x, y) option remember; convert(series(

%p `if`(n=0, 1, `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))

%p +b(n-1, y, x)*x + b(n-1, y, x+1)), z, 2), polynom)

%p end:

%p a:= n-> coeff(b(n, 0$2), z, 1):

%p seq(a(n), n=0..27);

%Y Column k=1 of A363493.

%Y Cf. A000110, A124419.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jun 06 2023