%I #35 Feb 25 2023 02:28:13

%S 0,1,15,204,3940,113865,4662231,256485040,18226108944,1623855701385,

%T 177195820499335,23237493232953516,3605437233380095620,

%U 653193551573628900289,136634950180317224866335,32681589590709963123092160,8863149183726257535369633856

%N Number of cycles in the complete bipartite graph K(n,n).

%C Also the number of chordless cycles in the n X n rook graph. - _Eric W. Weisstein_, Nov 27 2017

%H Andrew Howroyd, <a href="/A070968/b070968.txt">Table of n, a(n) for n = 1..100</a>

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

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

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

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

%F a(n) = Sum_{k=2..n} C(n,k)^2 * k! * (k-1)! / 2.

%F Recurrence: (n-2)^2*(2*n^3 - 19*n^2 + 58*n - 59)*a(n) = 2*(2*n^7 - 31*n^6 + 200*n^5 - 700*n^4 + 1442*n^3 - 1764*n^2 + 1205*n - 363)*a(n-1) - (n-1)^2*(2*n^7 - 35*n^6 + 266*n^5 - 1139*n^4 + 2962*n^3 - 4671*n^2 + 4130*n - 1578)*a(n-2) + 2*(n-2)^2*(n-1)^2*(2*n^5 - 26*n^4 + 134*n^3 - 342*n^2 + 431*n - 217)*a(n-3) - (n-3)^2*(n-2)^2*(n-1)^2*(2*n^3 - 13*n^2 + 26*n - 18)*a(n-4). - _Vaclav Kotesovec_, Mar 08 2016

%F a(n) ~ c * n! * (n-1)!, where c = BesselI(0,2)/2 = 1.1397926511680336337186... . - _Vaclav Kotesovec_, Mar 08 2016

%p seq(simplify((1/4)*hypergeom([1, 2, 2-n, 2-n], [3], 1)*(n-1)^2*n^2), n=1..20); # _Robert Israel_, Jan 09 2018

%t Table[Sum[Binomial[n, k]^2*k!*(k - 1)!, {k, 2, n}]/2, {n, 1, 17}]

%t Table[n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, 1] - 1)/2, {n, 20}] (* _Eric W. Weisstein_, Dec 14 2017 *)

%o (PARI) for(n=1,50,print1(sum(k=2,n,binomial(n,k)^2 * k! * (k-1)!/2),","))

%Y Row sums of A291909.

%Y Main diagonal of A360849.

%Y Cf. A002807, A068087, A288035.

%K nonn

%O 1,3

%A Sharon Sela (sharonsela(AT)hotmail.com), May 17 2002

%E More terms from _Benoit Cloitre_ and _Robert G. Wilson v_, May 20 2002

%E a(16)-a(17) from _Andrew Howroyd_, Jan 08 2018