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A070968 Number of cycles in the complete bipartite graph K(n,n).

%I A070968 #36 Feb 16 2025 08:32:46
%S A070968 0,1,15,204,3940,113865,4662231,256485040,18226108944,1623855701385,
%T A070968 177195820499335,23237493232953516,3605437233380095620,
%U A070968 653193551573628900289,136634950180317224866335,32681589590709963123092160,8863149183726257535369633856
%N A070968 Number of cycles in the complete bipartite graph K(n,n).
%C A070968 Also the number of chordless cycles in the n X n rook graph. - _Eric W. Weisstein_, Nov 27 2017
%H A070968 Andrew Howroyd, <a href="/A070968/b070968.txt">Table of n, a(n) for n = 1..100</a>
%H A070968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>
%H A070968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A070968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A070968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%F A070968 a(n) = Sum_{k=2..n} C(n,k)^2 * k! * (k-1)! / 2.
%F A070968 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 A070968 a(n) ~ c * n! * (n-1)!, where c = BesselI(0,2)/2 = 1.1397926511680336337186... . - _Vaclav Kotesovec_, Mar 08 2016
%p A070968 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 A070968 Table[Sum[Binomial[n, k]^2*k!*(k - 1)!, {k, 2, n}]/2, {n, 1, 17}]
%t A070968 Table[n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, 1] - 1)/2, {n, 20}] (* _Eric W. Weisstein_, Dec 14 2017 *)
%o A070968 (PARI) for(n=1,50,print1(sum(k=2,n,binomial(n,k)^2 * k! * (k-1)!/2),","))
%Y A070968 Row sums of A291909.
%Y A070968 Main diagonal of A360849.
%Y A070968 Cf. A002807, A068087, A288035.
%K A070968 nonn
%O A070968 1,3
%A A070968 Sharon Sela (sharonsela(AT)hotmail.com), May 17 2002
%E A070968 More terms from _Benoit Cloitre_ and _Robert G. Wilson v_, May 20 2002
%E A070968 a(16)-a(17) from _Andrew Howroyd_, Jan 08 2018