A212221 - OEIS

%I #26 Sep 21 2018 22:15:12

%S 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,

%T 0,1,138,2238,4780,1095,21,0,0,1,282,9546,46420,32955,2856,28,0,0,1,

%U 570,38958,385660,617775,168546,6412,36

%N 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).

%C 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.

%H Alois P. Heinz, <a href="/A212221/b212221.txt">Antidiagonals n = 1..100, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>

%F 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.

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

%e Square array A(n,k) begins:

%e    0,    0,     0,      0,       0,         0,          0, ...

%e    0,    0,     0,      0,       0,         0,          0, ...

%e    1,    1,     1,      1,       1,         1,          1, ...

%e    3,   12,    30,     66,     138,       282,        570, ...

%e    6,   78,   474,   2238,    9546,     38958,     155994, ...

%e   10,  340,  4780,  46420,  385660,   2995540,   22666780, ...

%e   15, 1095, 32955, 617775, 9248595, 123920295, 1569542955, ...

%p P:= proc(n) option remember;

%p       unapply(expand(add(add(Stirling2(n, k) *Stirling2(n, m)

%p        *mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=1..n), k=1..n)), q)

%p     end:

%p A:= (n, k)-> P(k)(n)/(2*n):

%p seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

%t 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 *)

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

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

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

%K nonn,tabl

%O 1,10

%A _Alois P. Heinz_, May 06 2012