This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A212221 #27 Feb 16 2025 08:33:17
%S A212221 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 A212221 0,1,138,2238,4780,1095,21,0,0,1,282,9546,46420,32955,2856,28,0,0,1,
%U A212221 570,38958,385660,617775,168546,6412,36
%N 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).
%C A212221 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 A212221 Alois P. Heinz, <a href="/A212221/b212221.txt">Antidiagonals n = 1..100, flattened</a>
%H A212221 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>
%H A212221 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%F A212221 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 A212221 A(n,n) = A282247(n).
%e A212221 Square array A(n,k) begins:
%e A212221 0, 0, 0, 0, 0, 0, 0, ...
%e A212221 0, 0, 0, 0, 0, 0, 0, ...
%e A212221 1, 1, 1, 1, 1, 1, 1, ...
%e A212221 3, 12, 30, 66, 138, 282, 570, ...
%e A212221 6, 78, 474, 2238, 9546, 38958, 155994, ...
%e A212221 10, 340, 4780, 46420, 385660, 2995540, 22666780, ...
%e A212221 15, 1095, 32955, 617775, 9248595, 123920295, 1569542955, ...
%p A212221 P:= proc(n) option remember;
%p A212221 unapply(expand(add(add(Stirling2(n, k) *Stirling2(n, m)
%p A212221 *mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=1..n), k=1..n)), q)
%p A212221 end:
%p A212221 A:= (n, k)-> P(k)(n)/(2*n):
%p A212221 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A212221 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 A212221 Rows 1+2,3-4 give: A000004, A000012, A089143(n-1) = 1/2*A182464(n-2) = 1/3*A182467(n-2).
%Y A212221 Columns 1-2 give: A000217(n-2), 1/(2*n)*A115400(n).
%Y A212221 Cf. A008277, A016777, A033428, A212220, A282247.
%K A212221 nonn,tabl
%O A212221 1,10
%A A212221 _Alois P. Heinz_, May 06 2012