A363454 - OEIS

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A363454

Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements and no block contains both odd and even elements.

1, 0, 1, 1, 2, 4, 11, 28, 87, 266, 952, 3381, 13513, 53915, 237113, 1046732, 5016728, 24186664, 125121009, 652084528, 3615047527, 20211789423, 119384499720, 711572380960, 4455637803543, 28162688795697, 186152008588691, 1242276416218540, 8636436319397292

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OFFSET

0,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..650

Wikipedia, Partition of a set

FORMULA

a(n) = Sum_{k=0..floor(n/2)} Stirling2(floor(n/2),k) * Stirling2(ceiling(n/2),k).

a(2n) = A047797(n).

EXAMPLE

a(0) = 1: () the empty partition.

a(1) = 0.

a(2) = 1: 1|2.

a(3) = 1: 13|2.

a(4) = 2: 13|24, 1|2|3|4.

a(5) = 4: 135|24, 13|2|4|5, 15|2|3|4, 1|2|35|4.

a(6) = 11: 135|246, 13|24|5|6, 13|26|4|5, 13|2|46|5, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46, 1|2|3|4|5|6.

a(7) = 28: 1357|246, 135|24|6|7, 137|24|5|6, 13|24|57|6, 135|26|4|7, 135|2|46|7, 137|26|4|5, 13|26|4|57, 137|2|46|5, 13|2|46|57, 13|2|4|5|6|7, 157|24|3|6, 15|24|37|6, 17|24|35|6, 1|24|357|6, 157|26|3|4, 15|26|37|4, 157|2|3|46, 15|2|37|46, 15|2|3|4|6|7, 17|26|35|4, 1|26|357|4, 17|2|35|46, 1|2|357|46, 1|2|35|4|6|7, 17|2|3|4|5|6, 1|2|37|4|5|6, 1|2|3|4|57|6.

MAPLE

a:= n-> (h-> add(Stirling2(h, k)*Stirling2(n-h, k), k=0..h))(iquo(n, 2)):

seq(a(n), n=0..40);

# second Maple program:

b:= proc(n, x, y) option remember; `if`(abs(x-y)>n, 0,

`if`(n=0, 1, `if`(x>0, b(n-1, y, x)*x, 0)+b(n-1, y, x+1)))

end:

a:= n-> b(n, 0$2):

seq(a(n), n=0..40);

CROSSREFS

Bisection gives A047797 (even part).

Cf. A000110, A124419, A124425, A363451.