A007079 Number of labeled regular tournaments with 2n+1 nodes.

1, 2, 24, 2640, 3230080, 48251508480, 9307700611292160, 24061983498249428379648, 855847205541481495117975879680, 427102683126284520201657800159366676480, 3035991776725501434069099002640396043332019814400, 311112533558482034321687955029997989477274014274150137856000

Offset: 0

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Brendan McKay, Table of n, a(n) for n = 0..17
• Mikhail Isaev, Brendan D. McKay, and Rui-Ray Zhang, Cumulant expansion for counting Eulerian orientations, arXiv:2309.15473 [math.CO], 2023. See Table 1 at p. 40.
• Brendan D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221.
• Brendan D. McKay, The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs, Combinatorica 10 (1990), 367-377.
• Index entries for sequences related to tournaments

Programs

Mathematica

(* This program is not convenient for more than 5 terms *)
a[n_] := (xx = Sequence @@ Table[ {x[k], 0, n}, {k, 1, 2*n + 1}]; Coefficient[ Normal @ Series[ Product[x[j] + x[k], {j, 1, (2*n + 1) - 1}, {k, j + 1, (2*n + 1)}], xx], Product[x[j] , {j, 1, (2*n + 1)}]^(((2*n + 1) - 1)/2)]); a[0] = 1; Table[a[n], {n, 0, 4}] (* Jean-François Alcover, Apr 10 2013 *)

PARI

/* not convenient for more than 5 terms: */
sym(k)=eval(Str("x" k));
pr(n)=prod(j=1,n-1, prod(k=j+1, n, sym(j) + sym(k) ) );
a(n)=
{
    my( p = pr(2*n+1) );
    for (k=1, 2*n+1, p = polcoeff(p, n, sym(k) );  );
    return( p );
} \\ Joerg Arndt, Apr 10 2013

PARI

a(n)={ local(M=Map());
my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
my(recurse(p, i, q, v, e)=if(e<=n, if(i<0, acc(x^e+q, v), my(t=polcoeff(p, i)); for(k=0, if(i==n, 0, t), self()(p, i-1, (t-k+x*k)*x^i+q, binomial(t, k)*v, e+t-k)))));
my(iterate(v, k, f)=for(i=1, k, v=f(v)); v);
iterate(Mat([1, 1]), 2*n, src->M=Map(); for(i=1, matsize(src)[1], my(p=src[i, 1]); recurse(p, poldegree(p), 0, src[i, 2], 0)); Mat(M))[1,2]
} \\ Andrew Howroyd, Jan 08 2018

Formula

a(n) is the coefficient of (x1 x2 ... xn)^((n-1)/2) in (x1+x2)(x1+x3)...(x(n-1)+xn). - Jim Ferry (ferry(AT)metsci.com), Sep 29 2005

Extensions

a(11) from Andrew Howroyd, Jan 08 2018