A291909 Triangle read by rows: T(n,k) is the coefficient of x^(2*k) in the cycle polynomial of the complete bipartite graph K_{n,n}, 1 <= k <= n.

0, 0, 1, 0, 9, 6, 0, 36, 96, 72, 0, 100, 600, 1800, 1440, 0, 225, 2400, 16200, 51840, 43200, 0, 441, 7350, 88200, 635040, 2116800, 1814400, 0, 784, 18816, 352800, 4515840, 33868800, 116121600, 101606400, 0, 1296, 42336, 1143072, 22861440, 304819200, 2351462400, 8230118400, 7315660800

Offset

1,5

Comments

Also the coefficients of x^(2*k) in the chordless cycle polynomial of the n X n rook graph. - Eric W. Weisstein, Feb 21 2018

Links

Pontus von Brömssen, Rows n = 1..100, flattened (rows n = 1..60 from Vincenzo Librandi).

Eric Weisstein's World of Mathematics, Chordless Cycle

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Eric Weisstein's World of Mathematics, Cycle Polynomial

Formula

T(n, k) = binomial(n, k)^2*k!*(k - 1)!/2 for k > 1.

Example

Cycle polynomials are
        0
      x^4
    9 x^4 +   6 x^6
   36 x^4 +  96 x^6 +   72 x^8
  100 x^4 + 600 x^6 + 1800 x^8 + 1440 x^10
  ...
so the first few rows are
  0;
  0,  1;
  0,  9,  6;
  0, 36, 96, 72;
  ...

Mathematica

CoefficientList[Table[Sum[Binomial[n, k]^2 k! (k - 1)! x^k, {k, 2, n}]/2, {n, 10}], x] // Flatten
Join[{0}, CoefficientList[Table[n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, x] - 1)/2, {n, 2, 10}], x]] // Flatten (* Eric W. Weisstein, Feb 21 2018 *)

Prog

(PARI) T(n, k) = if(k>1, binomial(n, k)^2*k!*(k - 1)!/2, 0) \\ Andrew Howroyd, Apr 29 2018

Crossrefs

Cf. A070968 (row sums), A010796 (main diagonal).

Sequence in context: A197413 A021055 A199067 * A011114 A298243 A020783

Adjacent sequences: A291906 A291907 A291908 * A291910 A291911 A291912

Keyword

nonn,tabl

Author

Eric W. Weisstein, Sep 05 2017

Extensions

Terms T(n,0) for n >= 3 deleted (in order to have a regular triangle) by Pontus von Brömssen, Sep 06 2022

Status

approved