A322279 - OEIS

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 #15 Dec 03 2018 21:37:15

%S 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 132,618,390,0,0,1,7,30,300,3156,15990,6062,0,0,1,8,42,570,9980,

%U 136980,668526,134526,0,0,1,9,56,966,24330,616260,10015092,43558242,4172198,0,0

%N 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 Not all colors need to be used.

%H Andrew Howroyd, <a href="/A322279/b322279.txt">Table of n, a(n) for n = 0..1274</a>

%H 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 k-th column is the logarithmic transform of the k-th column of A322280.

%F E.g.f of k-th column: 1 + log(Sum_{n>=0} A322280(n,k)*x^n/n!).

%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 0 2 6 12 20 30 ...

%e 3 | 0 0 6 42 132 300 570 ...

%e 4 | 0 0 38 618 3156 9980 24330 ...

%e 5 | 0 0 390 15990 136980 616260 1956810 ...

%e 6 | 0 0 6062 668526 10015092 65814020 277164210 ...

%e 7 | 0 0 134526 43558242 1199364852 11878194300 67774951650 ...

%e ...

%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*2^binomial(j, 2)) + O(x*x^n));

%o my(W=Mat(vector(n, k, Col(serlaplace(1 + log(serconvol(q, p^k)))))));

%o matconcat([1, W]);

%o }

%o my(T=M(7)); for(n=1, #T, print(T[n,]))

%Y Columns k=2..5 are A002027, A002028, A002029, A002030.

%Y Cf. A058843, A058875, A322278, A322280.

%K nonn,tabl

%O 0,8

%A _Andrew Howroyd_, Dec 01 2018