A327369 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

1, 1, 0, 1, 0, 1, 2, 0, 6, 0, 15, 12, 30, 4, 3, 314, 320, 260, 80, 50, 0, 13757, 10890, 5445, 1860, 735, 66, 15, 1142968, 640836, 228564, 64680, 16800, 2772, 532, 0, 178281041, 68362504, 17288852, 3666600, 702030, 115416, 17892, 1016, 105

Offset

0,7

Links

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

Formula

Column-wise binomial transform of A327377.

E.g.f.: 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
      1     0
      1     0     1
      2     0     6     0
     15    12    30     4     3
    314   320   260    80    50     0
  13757 10890  5445  1860   735    66    15

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Count[Length/@Split[Sort[Join@@#]], 1]==k&]], {n, 0, 5}, {k, 0, n}]

Prog

(PARI)
Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.
  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(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(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 + 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 A006125.

Row sums without the first column are A245797.

Column k = 0 is A059167.

Column k = 1 is A277072.

Column k = 2 is A277073.

Column k = 3 is A277074.

Column k = n is A123023.

Column k = n - 1 is A327370.

The unlabeled version is A327371.

The covering version is A327377.

Cf. A004110, A055302, A055314, A059166, A059167, A100743, A327227, A327228, A327362, A327364.

Sequence in context: A285538 A293935 A285479 * A296620 A263789 A081153

Adjacent sequences: A327366 A327367 A327368 * A327370 A327371 A327372

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 04 2019

Extensions

Terms a(28) and beyond from Andrew Howroyd, Sep 09 2019

Status

approved