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A256031 Number of irreducible idempotents in partial Brauer monoid PB_n.

%I A256031 #26 Feb 02 2023 02:24:16
%S A256031 2,3,12,30,240,840,10080,45360,725760,3991680,79833600,518918400,
%T A256031 12454041600,93405312000,2615348736000,22230464256000,711374856192000,
%U A256031 6758061133824000,243290200817664000,2554547108585472000,102181884343418880000,1175091669949317120000
%N A256031 Number of irreducible idempotents in partial Brauer monoid PB_n.
%C A256031 Table 2 in chapter 7 of the preprint contains a typo: a(9) is not 725860. - _R. J. Mathar_, Mar 14 2015
%H A256031 I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, J. Hyde and N. Loughlin, <a href="http://arxiv.org/abs/1408.2021">Enumeration of idempotents in diagram semigroups and algebras</a>, arXiv preprint arXiv:1408.2021 [math.GR], 2014. See Prop. 22.
%F A256031 There are simple formulas for the two bisections - see Dolinka et al.
%F A256031 a(2n-1) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A208529(2n+1). - _Omar E. Pol_, Mar 14 2015
%F A256031 Sum_{n>=1} 1/a(n) = (e^2+3)/(4*e) = 1/e + sinh(1)/2. - _Amiram Eldar_, Feb 02 2023
%p A256031 A256031 := proc(n)
%p A256031     if type(n,'odd') then
%p A256031         2*n! ;
%p A256031     else
%p A256031         (n+1)*(n-1)! ;
%p A256031     end if;
%p A256031 end proc:
%p A256031 seq(A256031(n),n=1..20) ; # _R. J. Mathar_, Mar 14 2015
%t A256031 a[n_] := If[OddQ[n], 2*n!, (n + 1)*(n - 1)!];
%t A256031 Array[a, 20] (* _Jean-François Alcover_, Nov 24 2017, from Maple *)
%Y A256031 Cf. A052612, A052616, A052849, A098558, A208529, A256032, A174549 (bisection).
%K A256031 nonn
%O A256031 1,1
%A A256031 _N. J. A. Sloane_, Mar 14 2015