A327377 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

1, 0, 0, 0, 0, 1, 1, 0, 3, 0, 10, 12, 12, 4, 3, 253, 260, 160, 60, 35, 0, 12068, 9150, 4230, 1440, 480, 66, 15, 1052793, 570906, 195048, 53200, 12600, 2310, 427, 0, 169505868, 63523656, 15600032, 3197040, 585620, 95088, 14056, 1016, 105

Offset

0,9

Comments

A graph is covering if there are no isolated vertices.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Formula

Column-wise inverse binomial transform of A327369.

E.g.f.: exp(-x)*exp(x + U(x,y) + B(x*(1-y) + R(x,y))), where R(x,y) is the e.g.f. of A055302, U(x,y) is the e.g.f. of A055314 and B(x) + x is the e.g.f. of A059167. - Andrew Howroyd, Oct 05 2019

Example

Triangle begins:
      1
      0     0
      0     0     1
      1     0     3     0
     10    12    12     4     3
    253   260   160    60    35     0
  12068  9150  4230  1440   480    66    15

Prog

(PARI)
Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.
  my(U=sum(n=2, n, x^n*sum(k=1, n, stirling(n-2, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(R=sum(n=1, n, x^n*sum(k=1, n, stirling(n-1, n-k, 2)*y^k/k!)) + O(x*x^n));
  my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
  my(A=exp(-x + O(x*x^n))*exp(x + U + subst(B-x, x, x*(1-y) + R)));
  Vecrev(n!*polcoef(A, n), n + 1);
}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Oct 05 2019

Crossrefs

Row sums are A006129.

Column k = 0 is A100743.

Column k = n is A123023.

Row sums without the first column are A327227.

The non-covering version is A327369.

The unlabeled version is A327372.

Cf. A004110, A006125, A055302, A055314, A059167, A245797, A327362, A327364, A327370.

Sequence in context: A028852 A319202 A327072 * A095200 A090460 A071983

Adjacent sequences: A327374 A327375 A327376 * A327378 A327379 A327380

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 05 2019

Extensions

Terms a(28) and beyond from Andrew Howroyd, Oct 05 2019

Status

approved