A370064 Triangle read by rows: T(n,k) is the number of simple connected graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

1, 1, 0, 1, 0, 0, 1, 3, 0, 0, 10, 16, 12, 0, 0, 238, 250, 180, 60, 0, 0, 11368, 8496, 4560, 1920, 360, 0, 0, 1014888, 540568, 211680, 75600, 21000, 2520, 0, 0, 166537616, 61672192, 17186624, 4663680, 1226400, 241920, 20160, 0, 0, 50680432112, 12608406288, 2416430016, 469336896, 98431200, 20109600, 2963520, 181440, 0, 0

Offset: 0

Author

Andrew Howroyd, Feb 23 2024

Keywords

• nonn
• tabl

Examples

			Triangle begins:
        1;
        1,      0;
        1,      0,      0;
        1,      3,      0,     0;
       10,     16,     12,     0,     0;
      238,    250,    180,    60,     0,    0;
    11368,   8496,   4560,  1920,   360,    0, 0;
  1014888, 540568, 211680, 75600, 21000, 2520, 0, 0;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Crossrefs

Columns k=0..3 are A013922(n>1), A013923, A013924, A013925.

Row sums are A001187.

Cf. A001710, A325111 (unlabeled version).

Programs

PARI

J(p, n)={my(u=Vecrev(p,1+n)); forstep(k=n, 1, -1, u[k] -= k*u[k+1]; u[k]/=n+1-k); u}
G(n)={log(x/serreverse(x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))))}
T(n)={my(v=Vec(serlaplace( 1 + ((y-1)*x + serreverse(x/((1-y) + y*exp(G(n)))))/y ))); vector(#v, n, J(v[n], n-1))}
{ my(A=T(7)); for(i=1, #A, print(A[i])) }

Formula

T(n, n-2) = n!/2 = A001710(n) for n >= 2.