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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. 8
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, 132, 618, 390, 0, 0, 1, 7, 30, 300, 3156, 15990, 6062, 0, 0, 1, 8, 42, 570, 9980, 136980, 668526, 134526, 0, 0, 1, 9, 56, 966, 24330, 616260, 10015092, 43558242, 4172198, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Not all colors need to be used.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1274

R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596.

FORMULA

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

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

EXAMPLE

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 0      2        6         12          20          30 ...

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

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

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

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

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

...

PROG

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

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

  matconcat([1, W]);

}

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

CROSSREFS

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

Cf. A058843, A058875, A322278, A322280.

Sequence in context: A217257 A217315 A217593 * A292861 A292133 A304482

Adjacent sequences:  A322276 A322277 A322278 * A322280 A322281 A322282

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Dec 01 2018

STATUS

approved

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Last modified November 19 19:09 EST 2019. Contains 329323 sequences. (Running on oeis4.)