A343088 Triangle read by rows: T(n,k) is the number of connected labeled graphs with n edges and k vertices, 1 <= k <= n+1.

1, 0, 1, 0, 0, 3, 0, 0, 1, 16, 0, 0, 0, 15, 125, 0, 0, 0, 6, 222, 1296, 0, 0, 0, 1, 205, 3660, 16807, 0, 0, 0, 0, 120, 5700, 68295, 262144, 0, 0, 0, 0, 45, 6165, 156555, 1436568, 4782969, 0, 0, 0, 0, 10, 4945, 258125, 4483360, 33779340, 100000000

Offset: 0

Author

Andrew Howroyd, Apr 14 2021

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0, 1;
  0, 0, 3;
  0, 0, 1, 16;
  0, 0, 0, 15, 125;
  0, 0, 0,  6, 222, 1296;
  0, 0, 0,  1, 205, 3660,  16807;
  0, 0, 0,  0, 120, 5700,  68295,  262144;
  0, 0, 0,  0,  45, 6165, 156555, 1436568, 4782969;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Main diagonal is A000272.

Subsequent diagonals give the number of connected labeled graphs with n nodes and n+k edges for k=0..11: A057500, A061540, A061541, A061542, A061543, A096117, A061544 A096150, A096224, A182294, A182295, A182371.

Row sums are A322137.

Column sums are A001187.

Cf. A054923 (unlabeled), A062734 (transpose), A290776 (multigraphs), A322147 (loops allowed), A331437 (series-reduced).

Programs

Mathematica

row[n_] := (SeriesCoefficient[#, {y, 0, n}]& /@ CoefficientList[ Log[Sum[x^k*(1+y)^Binomial[k, 2]/k!, {k, 0, n+1}]] + O[x]^(n+2), x]* Range[0, n+1]!) // Rest;
Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Aug 03 2022, after Andrew Howroyd *)

PARI

Row(n)={Vec(serlaplace(polcoef(log(O(x^2*x^n)+sum(k=0, n+1, x^k*(1 + y + O(y*y^n))^binomial(k, 2)/k!)), n, y)), -(n+1))}
{ for(n=0, 8, print(Row(n))) }