

A322278


Triangle read by rows: T(n,k) is the number of kcolored connected graphs on n labeled nodes up to permutation of the colors.


7



1, 0, 1, 0, 3, 4, 0, 19, 84, 38, 0, 195, 2470, 3140, 728, 0, 3031, 108390, 307390, 186360, 26704, 0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256, 0, 2086099, 726782784, 9030799218, 20784069600, 14401134944, 3463311488, 251548592
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OFFSET

1,5


COMMENTS

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with k parts. See A322064 for the definition of stable partition.


LINKS



FORMULA

T(n,k) = (1/k!)*Sum_{j=0..k} (1)^(kj)*binomial(k,j)*A322279(n,j).


EXAMPLE

Triangle begins:
1;
0, 1;
0, 3, 4;
0, 19, 84, 38;
0, 195, 2470, 3140, 728;
0, 3031, 108390, 307390, 186360, 26704;
0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256;
...


PROG

(PARI)
M(n, K=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=vector(K, k, Col(serlaplace(log(serconvol(q, p^k))))));
Mat(vector(K, k, sum(i=1, k, (1)^(ki)*binomial(k, i)*W[i])/k!));
}
my(T=M(7)); for(n=1, #T, print(T[n, 1..n]))


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



