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A274547 Number of set partitions of [n] with alternating parity of elements. 4

%I

%S 1,1,2,4,8,18,40,101,254,723,2064,6586,21143,74752,266078,1029983,

%T 4013425,16843526,71136112,321150717,1458636308,7038678613,

%U 34161890155,175261038904,904125989974,4909033438008,26795600521492,153376337926066,882391616100249

%N Number of set partitions of [n] with alternating parity of elements.

%H Alois P. Heinz, <a href="/A274547/b274547.txt">Table of n, a(n) for n = 0..36</a>

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

%F a(n) = Sum_{k=0..n} A274581(n,k).

%e a(5) = 18: 12345, 1234|5, 123|45, 123|4|5, 12|345, 12|34|5, 12|3|45, 12|3|4|5, 145|23, 1|2345, 1|234|5, 1|23|45, 1|23|4|5, 145|2|3, 1|2|345, 1|2|34|5, 1|2|3|45, 1|2|3|4|5.

%e a(6) = 40: 123456, 12345|6, 1234|56, 1234|5|6, 123|456, 123|45|6, 123|4|56, 123|4|5|6, 1256|34, 12|3456, 12|345|6, 12|34|56, 12|34|5|6, 1256|3|4, 12|3|456, 12|3|45|6, 12|3|4|56, 12|3|4|5|6, 145|236, 145|23|6, 1|23456, 1|2345|6, 1|234|56, 1|234|5|6, 1|23|456, 1|23|45|6, 1|23|4|56, 1|23|4|5|6, 145|2|36, 145|2|3|6, 1|256|34, 1|2|3456, 1|2|345|6, 1|2|34|56, 1|2|34|5|6, 1|256|3|4, 1|2|3|456, 1|2|3|45|6, 1|2|3|4|56, 1|2|3|4|5|6.

%p b:= proc(l, i, t) option remember; `if`(l=[], 1, add(`if`(l[j]=t,

%p b(subsop(j=[][], l), j, 1-t), 0), j=[1, $i..nops(l)]))

%p end:

%p a:= n-> b([seq(irem(i, 2), i=2..n)], 1, 0):

%p seq(a(n), n=0..25);

%t b[l_, i_, t_] := b[l, i, t] = If[l == {}, 1, Sum[If[l[[j]] == t, b[ReplacePart[l, j -> Sequence[]], j, 1-t], 0], {j, Prepend[Range[i, Length[l]], 1]}]]; a[n_] := b[Table[Mod[i, 2], {i, 2, n}], 1, 0]; Table[a[n], {n, 0, 25}] (* _Jean-Fran├žois Alcover_, Feb 15 2017, translated from Maple *)

%Y Row sums of A274581.

%Y Cf. A124419, A274310 (parities alternate within blocks).

%Y Column k=2 of A274859.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jun 27 2016

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Last modified February 26 14:41 EST 2021. Contains 341632 sequences. (Running on oeis4.)