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 A212220 #36 Feb 16 2025 08:33:17
%S A212220 1,1,-3,2,0,1,-12,58,-137,154,-64,0,1,-27,324,-2223,9414,-24879,39528,
%T A212220 -33966,11828,0,1,-48,1064,-14244,126936,-784788,3409590,-10329081,
%U A212220 21197804,-27779384,20648794,-6476644,0,1,-75,2650,-58100,878200,-9632440,78681510
%N A212220 Triangle T(n,k), n>=0, 0<=k<=3n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete tripartite graph K_(n,n,n), highest powers first.
%C A212220 The complete tripartite graph K_(n,n,n) has 3*n vertices and 3*n^2 = A033428(n) edges. The chromatic polynomial of K_(n,n,n) has 3*n+1 = A016777(n) coefficients.
%H A212220 Alois P. Heinz, <a href="/A212220/b212220.txt">Rows n = 0..58, flattened</a>
%H A212220 Anatol N. Kirillov, <a href="https://doi.org/10.3842/SIGMA.2016.002">On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials</a>, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
%H A212220 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>
%H A212220 Wikipedia, <a href="https://en.wikipedia.org/wiki/Acyclic_orientation">Acyclic orientation</a>
%H A212220 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%H A212220 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multipartite_graph">Multipartite graph</a>
%F A212220 T(n,k) = [q^(3*n-k)] Sum_{k,m=0..n} S2(n,k) * S2(n,m) * (q-k-m)^n * Product_{i=0..k+m-1} (q-i) with S2 = A008277.
%F A212220 Sum_{k=0..3n} (-1)^k * T(n,k) = A370961(n). - _Alois P. Heinz_, May 02 2024
%e A212220 2 example graphs: +-------------+
%e A212220 . | +-------+ |
%e A212220 . +-o---o---o |
%e A212220 . \ / \ / \ /
%e A212220 . X X X
%e A212220 . / \ / \ / \
%e A212220 . o---o---o +-o---o---o |
%e A212220 . +-------+ | +-------+ |
%e A212220 . +-------------+
%e A212220 Graph: K_(1,1,1) K_(2,2,2)
%e A212220 Vertices: 3 6
%e A212220 Edges: 3 12
%e A212220 The complete tripartite graph K_(1,1,1) is the cycle graph C_3 with chromatic polynomial q*(q-1)*(q-2) = q^3 -3*q^2 +2*q => [1, -3, 2, 0].
%e A212220 Triangle T(n,k) begins:
%e A212220 1;
%e A212220 1, -3, 2, 0;
%e A212220 1, -12, 58, -137, 154, -64, 0;
%e A212220 1, -27, 324, -2223, 9414, -24879, 39528, ...
%e A212220 1, -48, 1064, -14244, 126936, -784788, 3409590, ...
%e A212220 1, -75, 2650, -58100, 878200, -9632440, 78681510, ...
%e A212220 1, -108, 5562, -180585, 4123350, -70008186, 912054348, ...
%e A212220 ...
%p A212220 P:= proc(n) option remember;
%p A212220 expand(add(add(Stirling2(n, k) *Stirling2(n, m)
%p A212220 *mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=0..n), k=0..n))
%p A212220 end:
%p A212220 T:= n-> seq(coeff(P(n), q, 3*n-k), k=0..3*n):
%p A212220 seq(T(n), n=0..6);
%t A212220 P[n_] := P[n] = 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}]];
%t A212220 T[n_] := Table[Coefficient[P[n], q, 3*n - k], {k, 0, 3*n}];
%t A212220 Array[T, 6] // Flatten (* _Jean-François Alcover_, May 29 2018, from Maple *)
%Y A212220 Columns k=0-1 give: A000012, (-1)*A033428.
%Y A212220 Row sums and last elements of rows give: A000007.
%Y A212220 Row lengths give: A016777.
%Y A212220 Cf. A008277, A182553, A212221, A370961.
%K A212220 sign,tabf
%O A212220 0,3
%A A212220 _Alois P. Heinz_, May 06 2012
%E A212220 T(0,0)=1 prepended by _Alois P. Heinz_, May 02 2024