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

1,5

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

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

Formula

T(n,k) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*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)^(k-i)*binomial(k, i)*W[i])/k!));
}
my(T=M(7)); for(n=1, #T, print(T[n, 1..n]))

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.

Sequence in context: A078067 A192442 A009126 * A102222 A171657 A287696

Adjacent sequences: A322275 A322276 A322277 * A322279 A322280 A322281

Keyword

nonn,tabl

Author

Andrew Howroyd, Dec 01 2018

Status

approved