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A070968
Number of cycles in the complete bipartite graph K(n,n).
9
0, 1, 15, 204, 3940, 113865, 4662231, 256485040, 18226108944, 1623855701385, 177195820499335, 23237493232953516, 3605437233380095620, 653193551573628900289, 136634950180317224866335, 32681589590709963123092160, 8863149183726257535369633856
OFFSET
1,3
COMMENTS
Also the number of chordless cycles in the n X n rook graph. - Eric W. Weisstein, Nov 27 2017
LINKS
Eric Weisstein's World of Mathematics, Chordless Cycle
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Rook Graph
FORMULA
a(n) = Sum_{k=2..n} C(n,k)^2 * k! * (k-1)! / 2.
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
a(n) ~ c * n! * (n-1)!, where c = BesselI(0,2)/2 = 1.1397926511680336337186... . - Vaclav Kotesovec, Mar 08 2016
MAPLE
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
MATHEMATICA
Table[Sum[Binomial[n, k]^2*k!*(k - 1)!, {k, 2, n}]/2, {n, 1, 17}]
Table[n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, 1] - 1)/2, {n, 20}] (* Eric W. Weisstein, Dec 14 2017 *)
PROG
(PARI) for(n=1, 50, print1(sum(k=2, n, binomial(n, k)^2 * k! * (k-1)!/2), ", "))
CROSSREFS
Row sums of A291909.
Main diagonal of A360849.
Sequence in context: A216465 A215903 A373294 * A075280 A093747 A061637
KEYWORD
nonn
AUTHOR
Sharon Sela (sharonsela(AT)hotmail.com), May 17 2002
EXTENSIONS
More terms from Benoit Cloitre and Robert G. Wilson v, May 20 2002
a(16)-a(17) from Andrew Howroyd, Jan 08 2018
STATUS
approved