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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. 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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 15 17:03 EST 2019. Contains 330000 sequences. (Running on oeis4.)