A304222 Triangle T(n,k) read by rows: number of simple connected graphs with n nodes and k endpoints, n >= 0, 0 <= k <= n.

1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 3, 1, 1, 1, 0, 11, 5, 3, 1, 1, 0, 61, 29, 14, 5, 2, 1, 0, 507, 224, 86, 25, 8, 2, 1, 0, 7442, 2666, 762, 184, 48, 11, 3, 1, 0, 197772, 50779, 10173, 1890, 374, 72, 16, 3, 1, 0, 9808209, 1653431, 220627, 29252, 4252, 660, 115, 20, 4, 1, 0

Offset: 0

Author

R. J. Mathar, May 11 2018

Keywords

• nonn
• tabl

Comments

Endpoints are vertices with 0 or 1 (less than 2) edges.

Examples

			The triangle starts in row n=0 with column 0 <= k <= n as:
       1;
       1,     0;
       0,     0,     1;
       1,     0,     1,    0;
       3,     1,     1,    1,   0;
      11,     5,     3,    1,   1,  0;
      61,    29,    14,    5,   2,  1,  0;
     507,   224,    86,   25,   8,  2,  1, 0;
    7442,  2666,   762,  184,  48, 11,  3, 1, 0;
  197772, 50779, 10173, 1890, 374, 72, 16, 3, 1, 0;
		

Links

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

Crossrefs

Cf. A001349 (row sums), A004108 (first column), A055290 (trees only), A327371.

Programs

PARI

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i) )}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
G(n)={sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 2^edges(p) * prod(i=1, #p, (1 - x^p[i])/(1 - (x*y)^p[i]) + O(x*x^(n-k)))); x^k*s/k!)}
T(n)={my(v=InvEulerMT(Vec(G(n)-1))); v[2]=y^2; concat([[1]], vector(#v, n, Vecrev(v[n], n+1))) }
my(A=T(10)); for(n=1, #A, print(A[n])) \\ Andrew Howroyd, Jan 22 2021

Extensions

Terms a(55) and beyond from Andrew Howroyd, Jan 22 2021