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A322280
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Array read by antidiagonals: T(n,k) is the number of graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.
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12
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 26, 1, 0, 1, 5, 28, 123, 162, 1, 0, 1, 6, 45, 340, 1635, 1442, 1, 0, 1, 7, 66, 725, 7108, 35043, 18306, 1, 0, 1, 8, 91, 1326, 20805, 254404, 1206915, 330626, 1, 0, 1, 9, 120, 2191, 48486, 1058885, 15531268, 66622083, 8488962, 1, 0
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OFFSET
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0,8
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COMMENTS
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Not all colors need to be used.
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LINKS
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FORMULA
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T(n,k) = n!*2^binomial(n,2) * [x^n](Sum_{i>=0} x^i/(i!*2^binomial(i,2)))^k.
T(n,k) = Sum_{j=0..k} binomial(k,j)*j!*A058843(n,j).
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EXAMPLE
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Array begins:
===============================================================
n\k| 0 1 2 3 4 5 6
---+-----------------------------------------------------------
0 | 1 1 1 1 1 1 1 ...
1 | 0 1 2 3 4 5 6 ...
2 | 0 1 6 15 28 45 66 ...
3 | 0 1 26 123 340 725 1326 ...
4 | 0 1 162 1635 7108 20805 48486 ...
5 | 0 1 1442 35043 254404 1058885 3216486 ...
6 | 0 1 18306 1206915 15531268 95261445 386056326 ...
7 | 0 1 330626 66622083 1613235460 15110296325 83645197446 ...
...
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MATHEMATICA
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nmax = 10;
T[n_, k_] := n!*2^Binomial[n, 2]*SeriesCoefficient[Sum[ x^i/(i!* 2^Binomial[i, 2]), {i, 0, nmax}]^k, {x, 0, n}];
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PROG
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(PARI)
M(n)={
my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
my(q=sum(j=0, n, x^j*j!*2^binomial(j, 2)) + O(x*x^n));
matconcat([1, Mat(vector(n, k, Col(serconvol(q, p^k))))]);
}
my(T=M(7)); for(n=1, #T, print(T[n, ]))
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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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