A331438 - OEIS

%I #17 Jan 24 2020 16:05:43

%S 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,

%T 120,180,420,700,837,765,395,105,15,1,0,0,0,0,0,0,427,1260,3780,10850,

%U 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

%N 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.

%H Andrew Howroyd, <a href="/A331438/b331438.txt">Table of n, a(n) for n = 1..1350</a> (first 20 rows)

%H D. M. Jackson and J. W. Reilly, <a href="https://doi.org/10.1016/0095-8956(75)90090-8">The enumeration of homeomorphically irreducible labeled graphs</a>, J. Combin. Theory, B 19 (1975), 272-286. See Table III.

%e Triangle begins:

%e 1,

%e 0,1,

%e 0,0,0,0,

%e 0,0,0,4,0,0,1,

%e 0,0,0,0,5,0,0,20,15,10,1,

%e 0,0,0,0,0,96,120,180,420,700,837,765,395,105,15,1,

%e 0,0,0,0,0,0,427,1260,3780,10850,24045,44814,68040,80955,70500,43232,18774,5880,1330,210,21,1,

%e 0,0,0,0,0,0,0,6448,23520,79800,269360,782880,1956136,4203360,7610340,11365676,...,

%e ...

%o (PARI) \\ See Jackson & Reilly for e.g.f.

%o 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!)}

%o Row(n)={Vecrev(n!*polcoef(log(H(n,y)), n), binomial(n,2)+1)}

%o { for(n=1, 6, print(Row(n))) } \\ _Andrew Howroyd_, Jan 24 2020

%Y Row sums are A003515.

%Y Cf. A060514, A331437 (transpose).

%K nonn,tabf

%O 1,11

%A _N. J. A. Sloane_, Jan 19 2020