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%I #28 May 16 2024 14:24:54
%S 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,
%T 45,340,1635,1442,1,0,1,7,66,725,7108,35043,18306,1,0,1,8,91,1326,
%U 20805,254404,1206915,330626,1,0,1,9,120,2191,48486,1058885,15531268,66622083,8488962,1,0
%N 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.
%C Not all colors need to be used.
%H Andrew Howroyd, <a href="/A322280/b322280.txt">Table of n, a(n) for n = 0..1274</a>
%H R. C. Read, <a href="http://cms.math.ca/10.4153/CJM-1960-035-0">The number of k-colored graphs on labelled nodes</a>, Canad. J. Math., 12 (1960), 410-414.
%H R. C. Read and E. M. Wright, <a href="http://dx.doi.org/10.4153/CJM-1970-066-1">Colored graphs: A correction and extension</a>, Canad. J. Math. 22 1970 594-596.
%F T(n,k) = n!*2^binomial(n,2) * [x^n](Sum_{i>=0} x^i/(i!*2^binomial(i,2)))^k.
%F T(n,k) = Sum_{j=0..k} binomial(k,j)*j!*A058843(n,j).
%e Array begins:
%e ===============================================================
%e n\k| 0 1 2 3 4 5 6
%e ---+-----------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 ...
%e 1 | 0 1 2 3 4 5 6 ...
%e 2 | 0 1 6 15 28 45 66 ...
%e 3 | 0 1 26 123 340 725 1326 ...
%e 4 | 0 1 162 1635 7108 20805 48486 ...
%e 5 | 0 1 1442 35043 254404 1058885 3216486 ...
%e 6 | 0 1 18306 1206915 15531268 95261445 386056326 ...
%e 7 | 0 1 330626 66622083 1613235460 15110296325 83645197446 ...
%e ...
%t nmax = 10;
%t T[n_, k_] := n!*2^Binomial[n, 2]*SeriesCoefficient[Sum[ x^i/(i!* 2^Binomial[i, 2]), {i, 0, nmax}]^k, {x, 0, n}];
%t Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Sep 23 2019 *)
%o (PARI)
%o M(n)={
%o my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
%o my(q=sum(j=0, n, x^j*j!*2^binomial(j, 2)) + O(x*x^n));
%o matconcat([1, Mat(vector(n, k, Col(serconvol(q, p^k))))]);
%o }
%o my(T=M(7)); for(n=1, #T, print(T[n,]))
%Y Columns k=0..4 are A000007, A000012, A047863, A191371, A223887.
%Y Main diagonal gives A372920.
%Y Cf. A058843, A058875, A322278, A322279.
%K nonn,tabl
%O 0,8
%A _Andrew Howroyd_, Dec 01 2018
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