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A212221
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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).
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3
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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, 0, 1, 138, 2238, 4780, 1095, 21, 0, 0, 1, 282, 9546, 46420, 32955, 2856, 28, 0, 0, 1, 570, 38958, 385660, 617775, 168546, 6412, 36
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OFFSET
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1,10
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, ...
3, 12, 30, 66, 138, 282, 570, ...
6, 78, 474, 2238, 9546, 38958, 155994, ...
10, 340, 4780, 46420, 385660, 2995540, 22666780, ...
15, 1095, 32955, 617775, 9248595, 123920295, 1569542955, ...
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MAPLE
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P:= proc(n) option remember;
unapply(expand(add(add(Stirling2(n, k) *Stirling2(n, m)
*mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=1..n), k=1..n)), q)
end:
A:= (n, k)-> P(k)(n)/(2*n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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