A212221 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is 1/(2*n) times the number of n-colorings of the complete tripartite graph K_(k,k,k).

0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 1, 12, 6, 0, 0, 1, 30, 78, 10, 0, 0, 1, 66, 474, 340, 15, 0, 0, 1, 138, 2238, 4780, 1095, 21, 0, 0, 1, 282, 9546, 46420, 32955, 2856, 28, 0, 0, 1, 570, 38958, 385660, 617775, 168546, 6412, 36

Offset: 1

Author

Alois P. Heinz, May 06 2012

Keywords

• nonn
• tabl

Comments

The complete tripartite graph K_(n,n,n) has 3*n vertices and 3*n^2 = A033428(n) edges; see A212220 for example. The chromatic polynomial of K_(n,n,n) has 3*n+1 = A016777(n) coefficients.

Examples

			Square array A(n,k) begins:
   0,    0,     0,      0,       0,         0,          0, ...
   0,    0,     0,      0,       0,         0,          0, ...
   1,    1,     1,      1,       1,         1,          1, ...
   3,   12,    30,     66,     138,       282,        570, ...
   6,   78,   474,   2238,    9546,     38958,     155994, ...
  10,  340,  4780,  46420,  385660,   2995540,   22666780, ...
  15, 1095, 32955, 617775, 9248595, 123920295, 1569542955, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..100, flattened
• Eric Weisstein's World of Mathematics, Complete Tripartite Graph
• Wikipedia, Chromatic Polynomial

Crossrefs

Rows 1+2,3-4 give: A000004, A000012, A089143(n-1) = 1/2*A182464(n-2) = 1/3*A182467(n-2).

Columns 1-2 give: A000217(n-2), 1/(2*n)*A115400(n).

Cf. A008277, A016777, A033428, A212220, A282247.

Programs

Maple

P:= proc(n) option remember;
      unapply(expand(add(add(Stirling2(n, k) *Stirling2(n, m)
       *mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=1..n), k=1..n)), q)
    end:
A:= (n, k)-> P(k)(n)/(2*n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

Mathematica

p[n_] := p[n] = Function[q, Expand[Sum[Sum[StirlingS2[n, k] * StirlingS2[n, m] * Product[q-i, {i, 0, k+m-1}]*(q-k-m)^n, {m, 1, n}], {k, 1, n}]]]; a[n_, k_] := p[k][n]/(2*n); Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

Formula

A(n,k) = 1/(2*n) * Sum_{j,m=1..k} S2(k,j) * S2(k,m) * (n-j-m)^k * Product_{i=0..j+m-1} (n-i) with S2 = A008277.

A(n,n) = A282247(n).