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A342088 Triangle read by rows: T(n,k) is the number of n-colorings of the vertices of the k-dimensional cross polytope such that no two adjacent vertices have the same color. 0 <= k <= n.

%I A342088 #28 Feb 16 2025 08:34:01
%S A342088 1,1,1,1,4,2,1,9,18,6,1,16,84,96,24,1,25,260,780,600,120,1,36,630,
%T A342088 4080,7560,4320,720,1,49,1302,15330,61320,78120,35280,5040,1,64,2408,
%U A342088 45696,351120,913920,866880,322560,40320
%N A342088 Triangle read by rows: T(n,k) is the number of n-colorings of the vertices of the k-dimensional cross polytope such that no two adjacent vertices have the same color. 0 <= k <= n.
%H A342088 Peter Kagey, <a href="/A342088/b342088.txt">Rows n = 0..100, flattened</a>
%H A342088 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H A342088 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H A342088 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%H A342088 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a>
%F A342088 T(n,n) = n!.
%F A342088 T(n,k) = Sum_{i=0..2*k} A334279(k,i)*n^i.
%F A342088 T(n,k) = n*T(n-1,k-1) + n*(n-1)*T(n-2,k-1).
%F A342088 T(n,k) = Sum_{j=0..k} n!k!/((n-k-j)!(k-j)!j!).
%e A342088 Triangle begins:
%e A342088   n\k| 0   1     2      3       4       5       6       7      8
%e A342088   ---+----------------------------------------------------------
%e A342088    0 | 1
%e A342088    1 | 1,  1
%e A342088    2 | 1,  4,    2
%e A342088    3 | 1,  9,   18,     6
%e A342088    4 | 1, 16,   84,    96,     24
%e A342088    5 | 1, 25,  260,   780,    600,    120
%e A342088    6 | 1, 36,  630,  4080,   7560,   4320,    720
%e A342088    7 | 1, 49, 1302, 15330,  61320,  78120,  35280,   5040
%e A342088    8 | 1, 64, 2408, 45696, 351120, 913920, 866880, 322560, 40320
%t A342088 T[n_, k_] := Sum[n! k!/((n - k - j)! (k - j)! j!), {j, 0, k}]
%Y A342088 Cf. A000012 (k=0), A000290 (k=1), A091940 (k=2), A115400 (k=3), A334281 (k=4), A342073 (k=5), A342074 (k=6), A342075 (k=7).
%Y A342088 Cf. A334279.
%K A342088 nonn,tabl
%O A342088 0,5
%A A342088 _Peter Kagey_, Feb 27 2021