A331438 Irregular triangle read by rows: T(n,k) = number of homeomorphically irreducible connected labeled graphs with n vertices and k edges, n >= 1, 0 <= k <= n*(n-1)/2.

1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 5, 0, 0, 20, 15, 10, 1, 0, 0, 0, 0, 0, 96, 120, 180, 420, 700, 837, 765, 395, 105, 15, 1, 0, 0, 0, 0, 0, 0, 427, 1260, 3780, 10850, 24045, 44814, 68040, 80955, 70500, 43232, 18774, 5880, 1330, 210, 21, 1, 0, 0, 0, 0, 0, 0, 0, 6448, 23520, 79800, 269360, 782880, 1956136, 4203360, 7610340, 11365676

Offset: 1

Author

N. J. A. Sloane, Jan 19 2020

Keywords

• nonn
• tabf

Examples

			Triangle begins:
1,
0,1,
0,0,0,0,
0,0,0,4,0,0,1,
0,0,0,0,5,0,0,20,15,10,1,
0,0,0,0,0,96,120,180,420,700,837,765,395,105,15,1,
0,0,0,0,0,0,427,1260,3780,10850,24045,44814,68040,80955,70500,43232,18774,5880,1330,210,21,1,
0,0,0,0,0,0,0,6448,23520,79800,269360,782880,1956136,4203360,7610340,11365676,...,
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1350 (first 20 rows)
• D. M. Jackson and J. W. Reilly, The enumeration of homeomorphically irreducible labeled graphs, J. Combin. Theory, B 19 (1975), 272-286. See Table III.

Crossrefs

Row sums are A003515.

Cf. A060514, A331437 (transpose).

Programs

PARI

\\ See Jackson & Reilly for e.g.f.
H(n,y)={my(A=O(x*x^n)); (exp(y*x/2 - (y*x)^2/4 + A)/sqrt(1 + y*x + A))*sum(k=0, n, ((1 + y)*exp(-y^2*x/(1+y*x) + A))^binomial(k,2) * (x*exp((y^3*x^2 + A)/(2*(1 + y*x))))^k / k!)}
Row(n)={Vecrev(n!*polcoef(log(H(n,y)), n), binomial(n,2)+1)}
{ for(n=1, 6, print(Row(n))) } \\ Andrew Howroyd, Jan 24 2020