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
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