A297527 Number of maximum matchings in the complete tripartite graph K_n,n,n.

3, 8, 324, 1728, 216000, 1728000, 444528000, 4741632000, 2073989836800, 27653197824000, 18403203151872000, 294451250429952000, 277246884511973376000, 5175275177556836352000, 6549957646595371008000000, 139732429794034581504000000, 228835142526030632976384000000

Offset

1,1

Comments

For even n, a maximum matching will be a perfect matching. For odd n there will be one unmatched vertex. - Andrew Howroyd, Jan 01 2018

Links

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

Eric Weisstein's World of Mathematics, Complete Tripartite Graph

Eric Weisstein's World of Mathematics, Matching

Eric Weisstein's World of Mathematics, Maximum Independent Edge Set

Formula

a(n) = binomial(n, floor(n/2))^3 * floor(n/2)! * ceiling(n/2)!^2 * (2-(-1)^n). - Andrew Howroyd, Jan 01 2018

-8*(n+2)*(9*n^2+34*n+30)*(n+1)^3*a(n)+12*(n+2)*(n^2+5*n+5)*a(n+1)+(n+3)*(9*n^2+16*n+5)*a(n+2) = 0. - Eric W. Weisstein, Jan 06 2018

Mathematica

Table[Binomial[n, Floor[n/2]]^3 Floor[n/2]! Ceiling[n/2]!^2 (2 - (-1)^n), {n, 20}]
RecurrenceTable[{-8 (1 + n)^3 (2 + n) (30 + 34 n + 9 n^2) a[n] + 12 (2 + n) (5 + 5 n + n^2) a[1 + n] + (3 + n) (5 + 16 n + 9 n^2) a[2 + n] == 0, a[1] == 3, a[2] == 8}, a[n], {n, 20}]

Prog

(PARI) a(n)={if(n%2==0, binomial(n, n/2)^3*(n/2)!^3, 3*binomial(n, (n-1)/2)^3*((n+1)/2)!^2*((n-1)/2)!)} \\ Andrew Howroyd, Jan 01 2018

Crossrefs

Cf. A293075, A297008, A297487.

Sequence in context: A081466 A092592 A162185 * A268141 A354119 A278974

Adjacent sequences: A297524 A297525 A297526 * A297528 A297529 A297530

Keyword

nonn

Author

Eric W. Weisstein, Dec 31 2017

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 01 2018

Status

approved