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A274547 Number of set partitions of [n] with alternating parity of elements. 4
1, 1, 2, 4, 8, 18, 40, 101, 254, 723, 2064, 6586, 21143, 74752, 266078, 1029983, 4013425, 16843526, 71136112, 321150717, 1458636308, 7038678613, 34161890155, 175261038904, 904125989974, 4909033438008, 26795600521492, 153376337926066, 882391616100249 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..36

Wikipedia, Partition of a set

FORMULA

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

EXAMPLE

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.

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.

MAPLE

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

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

    end:

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

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

MATHEMATICA

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 *)

CROSSREFS

Row sums of A274581.

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

Column k=2 of A274859.

Sequence in context: A317787 A019231 A333865 * A112483 A151381 A306200

Adjacent sequences:  A274544 A274545 A274546 * A274548 A274549 A274550

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 27 2016

STATUS

approved

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Last modified January 22 03:32 EST 2021. Contains 340360 sequences. (Running on oeis4.)