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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.

%I A260884 #19 Mar 06 2020 03:13:25
%S A260884 1,-1,2,-1,-43,254,4157,-70981,-1310398,40933619,1087746617,
%T A260884 -43668096946,-1926040182823,74089958942999,6019198050509282,
%U A260884 -160614025972447321,-28821072878928603043,164386116957516330494,190177688405403480505877,5097857816569586800024019
%N 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.
%H A260884 Alois P. Heinz, <a href="/A260884/b260884.txt">Table of n, a(n) for n = 0..300</a>
%F A260884 E.g.f.: exp(1 - cosh(x)) (even powers only). - _Ilya Gutkovskiy_, Jan 27 2020
%e A260884 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.
%p A260884 b:= proc(n, t) option remember; `if`(n=0, 1-2*t, add(
%p A260884       b(n-2*j, 1-t)*binomial(n-1, 2*j-1), j=1..n/2))
%p A260884     end:
%p A260884 a:= n-> b(2*n, 0):
%p A260884 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jun 28 2016
%t A260884 Table[Sum[BellY[2 n, k, Mod[Range[2 n], 2] - 1], {k, 0, 2 n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o A260884 (Sage)
%o A260884 def ComplementaryBell(m, n):
%o A260884     shapes = ([x*m for x in p] for p in Partitions(n))
%o A260884     return sum((-1)^len(s)*SetPartitions(sum(s), s).cardinality() for s in shapes)
%o A260884 A260884 = lambda n: ComplementaryBell(2,n)
%o A260884 [A260884(n) for n in (0..19)]
%o A260884 (PARI) a(n) = {polcoef(serlaplace(exp(1 - cosh(x + O(x^(2*n + 1))))), 2*n)} \\ _Andrew Howroyd_, Jan 27 2020
%Y A260884 This is case m=2 of A260875. Case m=1 are the complementary Bell numbers A000587.
%K A260884 sign
%O A260884 0,3
%A A260884 _Peter Luschny_, Sep 18 2015