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A070968 Number of cycles in the complete bipartite graph K(n,n). 6
0, 1, 15, 204, 3940, 113865, 4662231, 256485040, 18226108944, 1623855701385, 177195820499335, 23237493232953516, 3605437233380095620, 653193551573628900289, 136634950180317224866335, 32681589590709963123092160, 8863149183726257535369633856 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Also the number of chordless cycles in the n X n rook graph. - Eric W. Weisstein, Nov 27 2017

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..100

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

Cf. A002807, A068087.

Sequence in context: A238992 A216465 A215903 * A075280 A093747 A061637

Adjacent sequences:  A070965 A070966 A070967 * A070969 A070970 A070971

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

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Last modified October 28 21:20 EDT 2020. Contains 338064 sequences. (Running on oeis4.)