A369932 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 2, 11, 9, 3, 0, 0, 0, 0, 1, 15, 32, 16, 4, 0, 0, 0, 0, 1, 12, 63, 76, 25, 5, 0, 0, 0, 0, 0, 8, 89, 234, 162, 39, 6, 0, 0, 0, 0, 0, 5, 97, 515, 730, 332, 60, 9

Offset

1,20

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Formula

T(n,k) = A123551(k,n) - A123551(k-1,n).

Example

Triangle begins:
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 1, 3,  2;
  0, 0, 0, 0, 3,  5,  2;
  0, 0, 0, 0, 2, 11,  9,  3;
  0, 0, 0, 0, 1, 15, 32, 16,  4;
  0, 0, 0, 0, 1, 12, 63, 76, 25, 5;
  ...

Prog

(PARI)
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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n) = {my(s=O(x*x^n)); sum(k=0, n, forpart(p=k, s+=permcount(p) * edges(p, w->1+y^w+O(y*y^n)) * x^k * prod(i=1, #p, 1-(y*x)^p[i], 1+O(x^(n-k+1))) / k!)); s*(1-x)}
T(n)={my(r=Vec(substvec(G(n), [x, y], [y, x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y), i)) }
{ my(A=T(12)); for(i=1, #A, print(A[i])) }

Crossrefs

Row sums are A369290.

Column sums are A261919.

Main diagonal is A008483.

Cf. A342557 (connected), A123551 (without endpoints).

Sequence in context: A269162 A033909 A292240 * A324881 A350734 A305930

Adjacent sequences: A369929 A369930 A369931 * A369933 A369934 A369935

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 07 2024

Status

approved