A123534 Triangular array T(n,k) giving number of 2-connected graphs with n labeled nodes and k edges (n >= 3, n <= k <= n(n-1)/2).

1, 3, 6, 1, 12, 70, 100, 45, 10, 1, 60, 720, 2445, 3535, 2697, 1335, 455, 105, 15, 1, 360, 7560, 46830, 133581, 216951, 232820, 183540, 111765, 53627, 20307, 5985, 1330, 210, 21, 1, 2520, 84000, 835800, 3940440, 10908688, 20317528

Offset

3,2

References

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

Links

Andrew Howroyd, Rows 3 through 20, flattened (first 15 rows from R. W. Robinson)

Example

Triangle begins (n >= 3, k >= n):
  n
  3 | 1;
  4 | 3, 6, 1;
  5 | 12, 70, 100, 45, 10, 1;
  6 | 60, 720, 2445, 3535, 2697, 1335, 455, 105, 15, 1;
  ...

Mathematica

row[n_] := row[n] = Module[{s}, s = (n-1)!*Log[x/InverseSeries[#, x]& @ (x*D[#, x]& @ Log[Sum[(1+y)^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1) ])]; CoefficientList[Coefficient[s, x, n-1]/y^n, y]];
Table[row[n], {n, 3, 15}] // Flatten (* Jean-François Alcover, Aug 13 2019, after Andrew Howroyd *)

Prog

(PARI) row(n)={Vecrev((n-1)!*polcoef(log(x/serreverse(x*deriv(log(sum(k=0, n, (1 + y)^binomial(k, 2) * x^k / k!) + O(x*x^n))))), n-1)/y^n)}
{ for(n=3, 7, print(row(n))) } \\ Andrew Howroyd, Nov 30 2018

Crossrefs

Row sums give A013922.

Cf. A062734, A123527, A322139.

Sequence in context: A120229 A266151 A192100 * A100960 A130852 A228335

Adjacent sequences: A123531 A123532 A123533 * A123535 A123536 A123537

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 13 2006

Status

approved