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A322280 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. 7
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, 45, 340, 1635, 1442, 1, 0, 1, 7, 66, 725, 7108, 35043, 18306, 1, 0, 1, 8, 91, 1326, 20805, 254404, 1206915, 330626, 1, 0, 1, 9, 120, 2191, 48486, 1058885, 15531268, 66622083, 8488962, 1, 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, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.

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

FORMULA

T(n,k) = n!*2^binomial(n,2) * [x^n](Sum_{i>=0} x^i/(i!*2^binomial(i,2)))^k.

T(n,k) = Sum_{j=0..k} binomial(k,j)*j!*A058843(n,j).

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 1      6       15         28          45          66 ...

3  | 0 1     26      123        340         725        1326 ...

4  | 0 1    162     1635       7108       20805       48486 ...

5  | 0 1   1442    35043     254404     1058885     3216486 ...

6  | 0 1  18306  1206915   15531268    95261445   386056326 ...

7  | 0 1 330626 66622083 1613235460 15110296325 83645197446 ...

...

MATHEMATICA

nmax = 10;

T[n_, k_] := n!*2^Binomial[n, 2]*SeriesCoefficient[Sum[ x^i/(i!* 2^Binomial[i, 2]), {i, 0, nmax}]^k, {x, 0, n}];

Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Sep 23 2019 *)

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

  matconcat([1, Mat(vector(n, k, Col(serconvol(q, p^k))))]);

}

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

CROSSREFS

Columns k=2..4 are A047863, A191371, A223887.

Cf. A058843, A058875, A322278, A322279.

Sequence in context: A183135 A294042 A287316 * A210472 A320080 A246106

Adjacent sequences:  A322277 A322278 A322279 * A322281 A322282 A322283

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Dec 01 2018

STATUS

approved

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Last modified December 9 08:20 EST 2019. Contains 329877 sequences. (Running on oeis4.)