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A322279 Array read by antidiagonals: T(n,k) is the number of connected graphs on n labeled nodes, each node being colored with one of k colors, where no edge connects two nodes of the same color.

%I A322279 #15 Dec 03 2018 21:37:15
%S A322279 1,1,0,1,1,0,1,2,0,0,1,3,2,0,0,1,4,6,6,0,0,1,5,12,42,38,0,0,1,6,20,
%T A322279 132,618,390,0,0,1,7,30,300,3156,15990,6062,0,0,1,8,42,570,9980,
%U A322279 136980,668526,134526,0,0,1,9,56,966,24330,616260,10015092,43558242,4172198,0,0
%N A322279 Array read by antidiagonals: T(n,k) is the number of connected 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 A322279 Not all colors need to be used.
%H A322279 Andrew Howroyd, <a href="/A322279/b322279.txt">Table of n, a(n) for n = 0..1274</a>
%H A322279 R. C. Read, 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 A322279 k-th column is the logarithmic transform of the k-th column of A322280.
%F A322279 E.g.f of k-th column: 1 + log(Sum_{n>=0} A322280(n,k)*x^n/n!).
%e A322279 Array begins:
%e A322279 ===============================================================
%e A322279 n\k| 0 1      2        3          4           5           6
%e A322279 ---+-----------------------------------------------------------
%e A322279 0  | 1 1      1        1          1           1           1 ...
%e A322279 1  | 0 1      2        3          4           5           6 ...
%e A322279 2  | 0 0      2        6         12          20          30 ...
%e A322279 3  | 0 0      6       42        132         300         570 ...
%e A322279 4  | 0 0     38      618       3156        9980       24330 ...
%e A322279 5  | 0 0    390    15990     136980      616260     1956810 ...
%e A322279 6  | 0 0   6062   668526   10015092    65814020   277164210 ...
%e A322279 7  | 0 0 134526 43558242 1199364852 11878194300 67774951650 ...
%e A322279 ...
%o A322279 (PARI)
%o A322279 M(n)={
%o A322279   my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
%o A322279   my(q=sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n));
%o A322279   my(W=Mat(vector(n, k, Col(serlaplace(1 + log(serconvol(q, p^k)))))));
%o A322279   matconcat([1, W]);
%o A322279 }
%o A322279 my(T=M(7)); for(n=1, #T, print(T[n,]))
%Y A322279 Columns k=2..5 are A002027, A002028, A002029, A002030.
%Y A322279 Cf. A058843, A058875, A322278, A322280.
%K A322279 nonn,tabl
%O A322279 0,8
%A A322279 _Andrew Howroyd_, Dec 01 2018