A052616 -id:A052616 - cp's OEIS frontend

A256031 Number of irreducible idempotents in partial Brauer monoid PB_n.

2, 3, 12, 30, 240, 840, 10080, 45360, 725760, 3991680, 79833600, 518918400, 12454041600, 93405312000, 2615348736000, 22230464256000, 711374856192000, 6758061133824000, 243290200817664000, 2554547108585472000, 102181884343418880000, 1175091669949317120000

Offset: 1

N. J. A. Sloane, Mar 14 2015

nonn

Table 2 in chapter 7 of the preprint contains a typo: a(9) is not 725860. - R. J. Mathar, Mar 14 2015

I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, J. Hyde and N. Loughlin, Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014. See Prop. 22.

Cf. A052612, A052616, A052849, A098558, A208529, A256032, A174549 (bisection).

A256031 := proc(n)
    if type(n,'odd') then
        2*n! ;
    else
        (n+1)*(n-1)! ;
    end if;
end proc:
seq(A256031(n),n=1..20) ; # R. J. Mathar, Mar 14 2015

a[n_] := If[OddQ[n], 2*n!, (n + 1)*(n - 1)!];
Array[a, 20] (* Jean-François Alcover, Nov 24 2017, from Maple *)

There are simple formulas for the two bisections - see Dolinka et al.
a(2n-1) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A208529(2n+1). - Omar E. Pol, Mar 14 2015
Sum_{n>=1} 1/a(n) = (e^2+3)/(4*e) = 1/e + sinh(1)/2. - Amiram Eldar, Feb 02 2023

A256881 a(n) = n!/ceiling(n/2).

Original entry on oeis.org

1, 2, 3, 12, 40, 240, 1260, 10080, 72576, 725760, 6652800, 79833600, 889574400, 12454041600, 163459296000, 2615348736000, 39520825344000, 711374856192000, 12164510040883200, 243290200817664000, 4644631106519040000, 102181884343418880000, 2154334728240414720000

Offset: 1

M. F. Hasler, Apr 22 2015

nonn

Original name was: n!/round(n/2). - Robert Israel, Sep 03 2018

Robert Israel, Table of n, a(n) for n = 1..450
Pierre-Alain Sallard, Sum of repeated integrals of sinh.

Cf. A009445, A052612, A052616, A052849, A081457, A208529, A098558, A107991, A110468, A229244, A256031.

[Factorial(n)/Round(n/2): n in [1..30]]; // Vincenzo Librandi, Apr 23 2015

Maple
```
A256881 := n!/round(n/2);
```

Function[x, 1/x] /@
CoefficientList[Series[(Sinh[x] + x*Exp[x])/2, {x, 0, 20}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)

PARI
```
A256881(n)=n!/round(n/2)
```

a(2n) = 2*A009445(n) = A052612(2n-1) = A052616(2n-1) = A052849(2n-1) = A098558(2n-1) = A081457(3n-1) = A208529(2n+1) = A256031(2n-1).
a(2n+1) = A110468(n) = A107991(2n+2) = A229244(2n+1), n>=0.
From Robert Israel, Sep 03 2018: (Start)
E.g.f.: -(1+1/x)*log(1-x^2).
n*(n+1)*(n+2)*a(n)+(n+2)*a(n+1)-(n+3)*a(n+2)=0. (End)
a(n) = 2/([x^n](sinh(x) + x*exp(x))). - Pierre-Alain Sallard, Dec 15 2018
Sum_{n>=1} 1/a(n) = (3*e-1/e)/4 = (e + sinh(1))/2. - Amiram Eldar, Feb 02 2023

Definition clarified by Robert Israel, Sep 03 2018

cp's OEIS Frontend

A256031 Number of irreducible idempotents in partial Brauer monoid PB_n.

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