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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.
5
1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 84, 96, 24, 1, 25, 260, 780, 600, 120, 1, 36, 630, 4080, 7560, 4320, 720, 1, 49, 1302, 15330, 61320, 78120, 35280, 5040, 1, 64, 2408, 45696, 351120, 913920, 866880, 322560, 40320
OFFSET
0,5
LINKS
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Wikipedia, Cross-polytope
Wikipedia, TurĂ¡n graph
FORMULA
T(n,n) = n!.
T(n,k) = Sum_{i=0..2*k} A334279(k,i)*n^i.
T(n,k) = n*T(n-1,k-1) + n*(n-1)*T(n-2,k-1).
T(n,k) = Sum_{j=0..k} n!k!/((n-k-j)!(k-j)!j!).
EXAMPLE
Triangle begins:
n\k| 0 1 2 3 4 5 6 7 8
---+----------------------------------------------------------
0 | 1
1 | 1, 1
2 | 1, 4, 2
3 | 1, 9, 18, 6
4 | 1, 16, 84, 96, 24
5 | 1, 25, 260, 780, 600, 120
6 | 1, 36, 630, 4080, 7560, 4320, 720
7 | 1, 49, 1302, 15330, 61320, 78120, 35280, 5040
8 | 1, 64, 2408, 45696, 351120, 913920, 866880, 322560, 40320
MATHEMATICA
T[n_, k_] := Sum[n! k!/((n - k - j)! (k - j)! j!), {j, 0, k}]
CROSSREFS
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).
Cf. A334279.
Sequence in context: A259985 A144084 A021010 * A193607 A358735 A075397
KEYWORD
nonn,tabl
AUTHOR
Peter Kagey, Feb 27 2021
STATUS
approved