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.

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

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: A217593 A353434 A350529 * A350365 A331923 A342129

Adjacent sequences: A322276 A322277 A322278 * A322280 A322281 A322282

Keyword

nonn,tabl

Author

Andrew Howroyd, Dec 01 2018

Status

approved