A260884 Number of set partitions of a 2n-set into even blocks which have even length minus the number of partitions into even blocks which have odd length.

1, -1, 2, -1, -43, 254, 4157, -70981, -1310398, 40933619, 1087746617, -43668096946, -1926040182823, 74089958942999, 6019198050509282, -160614025972447321, -28821072878928603043, 164386116957516330494, 190177688405403480505877, 5097857816569586800024019

Offset: 0

Author

Peter Luschny, Sep 18 2015

Keywords

• sign

Examples

			a(5) = 254. Consider the shapes [10], [8, 2], [6, 4], [6, 2, 2], [4, 4, 2], [4, 2, 2, 2], [2, 2, 2, 2, 2]. Computing the number of associated set partitions gives -1 + 45 + 210 - 630 - 1575 + 3150 - 945 = 254.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

This is case m=2 of A260875. Case m=1 are the complementary Bell numbers A000587.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1-2*t, add(
      b(n-2*j, 1-t)*binomial(n-1, 2*j-1), j=1..n/2))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 28 2016

Mathematica

Table[Sum[BellY[2 n, k, Mod[Range[2 n], 2] - 1], {k, 0, 2 n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

PARI

a(n) = {polcoef(serlaplace(exp(1 - cosh(x + O(x^(2*n + 1))))), 2*n)} \\ Andrew Howroyd, Jan 27 2020

Sage

def ComplementaryBell(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return sum((-1)^len(s)*SetPartitions(sum(s), s).cardinality() for s in shapes)
A260884 = lambda n: ComplementaryBell(2,n)
[A260884(n) for n in (0..19)]

Formula

E.g.f.: exp(1 - cosh(x)) (even powers only). - Ilya Gutkovskiy, Jan 27 2020