A342557 T(n,m) is the number of unlabeled connected graphs without endpoints on m nodes with n edges, where T(n,m), m <= n, is a triangle read by rows.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 3, 5, 1, 0, 0, 0, 0, 2, 11, 8, 1, 0, 0, 0, 0, 1, 15, 31, 12, 1, 0, 0, 0, 0, 1, 12, 63, 71, 16, 1, 0, 0, 0, 0, 0, 8, 89, 231, 144, 21, 1, 0, 0, 0, 0, 0, 5, 97, 513, 707, 274, 27, 1

Offset: 1

Author

Hugo Pfoertner, May 21 2021

Keywords

• nonn
• tabl

Comments

The number of nonzero terms in the n-th row is A083920(n-1). - Hugo Pfoertner, Feb 01 2024

Examples

			The triangle begins
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 1, 3,  1;
  0, 0, 0, 0, 3,  5,  1;
  0, 0, 0, 0, 2, 11,  8,  1;
  0, 0, 0, 0, 1, 15, 31, 12,  1;
  0, 0, 0, 0, 1, 12, 63, 71, 16, 1;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50; first 24 rows from Hugo Pfoertner)

Crossrefs

Cf. A004108 (column sums), A342556 (row sums).

Cf. A083920, A369932 (not necessarily connected).

Programs

PARI

\\ Needs G() defined in A369932.
InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
T(n)={my(r=Vec(InvEulerMTS(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])) } \\ Andrew Howroyd, Feb 07 2024

Formula

Bivariate inverse Euler transform of A369932. - Andrew Howroyd, Feb 07 2024