A322278 Triangle read by rows: T(n,k) is the number of k-colored connected graphs on n labeled nodes up to permutation of the colors.

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

Offset: 1

Author

Andrew Howroyd, Dec 01 2018

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Row sums are A322064.

Columns k=2..4 are A001832(for n > 1), A322330, A322331.

Right diagonal is A001187.

Cf. A058843, A058875, A322279, A322280.

Programs

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)^(k-i)*binomial(k,i)*W[i])/k!));
}
my(T=M(7)); for(n=1, #T, print(T[n, 1..n]))

Formula

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