A293075 - OEIS

%I #13 Oct 04 2017 14:12:02

%S 4,51,1126,37201,1670136,96502339,6900041506,593717817921,

%T 60163621650316,7059439676098291,946047724677141054,

%U 143165355635117094481,24232437980331557100736,4550485215254864673978051,941387925046160753185319866,213240954445118902597065224449

%N Number of matchings in the complete tripartite graph K_{n,n,n}.

%H Andrew Howroyd, <a href="/A293075/b293075.txt">Table of n, a(n) for n = 1..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>

%F a(n) = Sum_{i,j,k} binomial(n, i)^2 * binomial(n, j) * binomial(n-i, j) * binomial(n-i, k) * binomial(n-j, k) * i!*j!*k!. - _Andrew Howroyd_, Oct 02 2017

%t Table[Sum[Binomial[n, i]^2 Binomial[n, j] Binomial[n - i, j] Binomial[n - i, k] Binomial[n - j, k] i! j! k!, {i, 0, n}, {j, 0, n - i}, {k, 0, Min[n - i, n - j]}], {n, 20}]

%t Table[Sum[(-1)^(i - n) Binomial[n, i]^2 Binomial[n, j] Binomial[-i + n, j] i! j! HypergeometricU[i - n, 1 + i - j, -1], {i, 0, n}, {j, 0, n - i}], {n, 20}]

%o (PARI) a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, min(n-i, n-j), binomial(n, i)^2 * binomial(n, j) * binomial(n-i, j) * binomial(n-i, k) * binomial(n-j, k) * i!*j!*k!))); \\ _Andrew Howroyd_, Oct 02 2017

%Y Cf. A002720 (matchings in complete bipartite graph).

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Sep 30 2017

%E Terms a(11) and beyond from _Andrew Howroyd_, Oct 02 2017