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

%I A007079 M2142 #36 Sep 30 2023 18:49:40
%S A007079 1,2,24,2640,3230080,48251508480,9307700611292160,
%T A007079 24061983498249428379648,855847205541481495117975879680,
%U A007079 427102683126284520201657800159366676480,3035991776725501434069099002640396043332019814400,311112533558482034321687955029997989477274014274150137856000
%N A007079 Number of labeled regular tournaments with 2n+1 nodes.
%D A007079 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007079 Brendan McKay, <a href="/A007079/b007079.txt">Table of n, a(n) for n = 0..17</a>
%H A007079 Mikhail Isaev, Brendan D. McKay, and Rui-Ray Zhang, <a href="https://arxiv.org/abs/2309.15473">Cumulant expansion for counting Eulerian orientations</a>, arXiv:2309.15473 [math.CO], 2023. See Table 1 at p. 40.
%H A007079 Brendan D. McKay, <a href="http://cs.anu.edu.au/~bdm/papers/LabelledEnumeration.pdf">Applications of a technique for labeled enumeration</a>, Congress. Numerantium, 40 (1983), 207-221.
%H A007079 Brendan D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/papers/rt.pdf">The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs</a>, Combinatorica 10 (1990), 367-377.
%H A007079 <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>
%F A007079 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
%t A007079 (* This program is not convenient for more than 5 terms *)
%t A007079 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 *)
%o A007079 (PARI) /* not convenient for more than 5 terms: */
%o A007079 sym(k)=eval(Str("x" k));
%o A007079 pr(n)=prod(j=1,n-1, prod(k=j+1, n, sym(j) + sym(k) ) );
%o A007079 a(n)=
%o A007079 {
%o A007079     my( p = pr(2*n+1) );
%o A007079     for (k=1, 2*n+1, p = polcoeff(p, n, sym(k) );  );
%o A007079     return( p );
%o A007079 } \\ _Joerg Arndt_, Apr 10 2013
%o A007079 (PARI)
%o A007079 a(n)={ local(M=Map());
%o A007079 my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
%o A007079 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)))));
%o A007079 my(iterate(v, k, f)=for(i=1, k, v=f(v)); v);
%o A007079 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]
%o A007079 } \\ _Andrew Howroyd_, Jan 08 2018
%K A007079 nonn,nice
%O A007079 0,2
%A A007079 _N. J. A. Sloane_, _Mira Bernstein_
%E A007079 a(11) from _Andrew Howroyd_, Jan 08 2018