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 and 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-François 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
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Dec 01 2018
STATUS
approved