A322330 -id:A322330 - cp's OEIS frontend

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

Andrew Howroyd, Dec 01 2018

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.

			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;
  ...

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

Row sums are A322064.
Columns k=2..4 are A001832(for n > 1), A322330, A322331.
Right diagonal is A001187.
Cf. A058843, A058875, A322279, A322280.

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]))

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

A322331 Number of 4-colored connected graphs on n labeled nodes up to permutation of the colors.

Original entry on oeis.org

38, 3140, 307390, 42747460, 9030799218, 3012940879620, 1628920258500230, 1451200592494754420, 2152262350514389189978, 5344908165470797467243700, 22297912999366719508496874990, 156537595118740106754291705258180, 1850935702258755131781978373277937218

Offset: 4

Andrew Howroyd, Dec 03 2018

nonn

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with 4 parts. See A322064 for the definition of stable partition.

Andrew Howroyd, Table of n, a(n) for n = 4..50

Column k=4 of A322278.
Cf. A058873 (not necessarily connected), A322064.
Cf. A001832, A322330.

\\ See A322278 for M.
{ my(N=20); M(N,4)[4..N, 4]~ }

cp's OEIS Frontend

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.

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