A291909 -id:A291909 - cp's OEIS frontend

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

0, 1, 15, 204, 3940, 113865, 4662231, 256485040, 18226108944, 1623855701385, 177195820499335, 23237493232953516, 3605437233380095620, 653193551573628900289, 136634950180317224866335, 32681589590709963123092160, 8863149183726257535369633856

Offset: 1

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

nonn

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

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

Row sums of A291909.
Main diagonal of A360849.
Cf. A002807, A068087, A288035.

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

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 *)

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

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

More terms from Benoit Cloitre and Robert G. Wilson v, May 20 2002

a(16)-a(17) from Andrew Howroyd, Jan 08 2018

A010796 a(n) = n!*(n+1)!/2.

Original entry on oeis.org

1, 6, 72, 1440, 43200, 1814400, 101606400, 7315660800, 658409472000, 72425041920000, 9560105533440000, 1491376463216640000, 271430516305428480000, 57000408424139980800000, 13680098021793595392000000, 3720986661927857946624000000

Offset: 1

N. J. A. Sloane

Column 2 in triangle A009963.
a(n) = A078740(n, 2), first column of (3, 2)-Stirling2 array.
Also the number of undirected Hamiltonian paths in the complete bipartite graph K_{n,n+1}. - Eric W. Weisstein, Sep 03 2017
Also, the number of undirected Hamiltonian cycles in the complete bipartite graph K_{n+1,n+1}. - Pontus von Brömssen, Sep 06 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Index entries for sequences related to factorial numbers.

Cf. A006472, A009963, A078740, A010790.

Main diagonal of A291909.

[Factorial(n)* Factorial(n+1) / 2: n in [1..20]]; // Vincenzo Librandi, Jun 11 2013

Table[n! (n + 1)! / 2, {n, 1, 20}] (* Vincenzo Librandi, Jun 11 2013 *)
Times@@@Partition[Range[20]!,2,1]/2 (* Harvey P. Dale, Jul 04 2017 *)

for(n=1,30, print1(n!*(n+1)!/2, ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = 2^(n-1) * A006472(n+1).
a(n) = A010790(n)/2.
E.g.f.: (hypergeom([1, 2], [], x)-1)/2.
a(n) = Product_{k=1..n-1} (k^2+3*k+2). - Gerry Martens, May 09 2016
E.g.f.: x*hypergeom([1, 3], [], x). - Robert Israel, May 09 2016
From Amiram Eldar, Jun 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(BesselI(1, 2) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - BesselJ(1, 2)). (End)

A296546 Triangle read by rows T(n,k): number of undirected cycles of length k in the complete tripartite graph K_{n,n,n} (n = 1...; k = 3..3n).

Original entry on oeis.org

1, 8, 15, 24, 16, 27, 108, 324, 774, 1620, 2268, 1584, 64, 396, 1728, 7200, 27648, 87480, 232704, 476928, 663552, 463104, 125, 1050, 6000, 35800, 198000, 977400, 4392000, 17068320, 56376000, 151632000, 311040000, 430272000, 299289600

Offset: 1

Eric W. Weisstein, Dec 15 2017

			Written as cycle polynomials:
  x^3
  8 x^3 + 15 x^4 + 24 x^5 + 16 x^6
  27 x^3 + 108 x^4 + 324 x^5 + 774 x^6 + 1620 x^7 + 2268 x^8 + 1584 x^9
  64 x^3 + 396 x^4 + 1728 x^5 + 7200 x^6 + 27648 x^7 + 87480 x^8 + 232704 x^9 + 476928 x^10 + 663552 x^11 + 463104 x^12
giving the array
  1
  8, 15, 24, 16
  27, 108, 324, 774, 1620, 2268, 1584
  64, 396, 1728, 7200, 27648, 87480, 232704, 476928, 663552, 463104

Eric W. Weisstein, Table of n, a(n) for n = 1..117 (rows n = 1..9, flattened).
Eric Weisstein's World of Mathematics, Cycle Polynomial.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph.

Cf. A234616 (number of undirected cycles in K_{n,n,n}).
Cf. A144151 (cycle polynomial coefficients of complete graph K_n).
Cf. A291909 (cycle polynomial coefficients of complete bipartite graph K_{n,n}).

Table[Tally[Length /@ FindCycle[CompleteGraph[{n, n, n}], Infinity, All]][[All, 2]], {n, 4}] // Flatten

Row sums of T(n,k) give A234616(n).

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

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A010796 a(n) = n!*(n+1)!/2.

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A296546 Triangle read by rows T(n,k): number of undirected cycles of length k in the complete tripartite graph K_{n,n,n} (n = 1...; k = 3..3n).

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