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

Author

Gus Wiseman, Sep 05 2019

Keywords

• nonn
• tabl

Comments

A graph is covering if there are no isolated vertices.

Examples

			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
		

Links

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

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.

Programs

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

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

Extensions

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