A350909 - OEIS

%I #7 Jan 29 2022 22:31:28

%S 1,0,2,0,0,12,6,0,0,0,128,186,108,24,0,0,0,0,2000,5640,7840,6540,3330,

%T 960,120,0,0,0,0,0,41472,189480,456720,730830,832370,690300,416160,

%U 178230,51480,9000,720,0,0,0,0,0,0,1075648,7178640,26035800,65339820

%N Triangle read by rows: T(n,k) is the number of weakly connected acyclic digraphs on n labeled nodes with k arcs, k=0..n*(n-1).

%H Andrew Howroyd, <a href="/A350909/b350909.txt">Table of n, a(n) for n = 1..1350</a> (rows 1..20)

%e Triangle begins:

%e   [1] 1;

%e   [2] 0, 2;

%e   [3] 0, 0, 12,   6;

%e   [4] 0, 0,  0, 128,  186,  108,   24;

%e   [5] 0, 0,  0,   0, 2000, 5640, 7840, 6540, 3330, 960, 120;

%e   ...

%o (PARI)

%o G(n)={my(v=vector(n+1)); v[1]=1; for(n=1, n, v[n+1]=sum(k=1, n, -(-1)^k*(1+y)^(k*(n-k))*v[n-k+1]/k!))/n!; Ser(v)}

%o row(n)={Vecrev(n!*polcoef(log(G(n)), n))}

%o { for(n=1, 6, print(row(n))) }

%Y Row sums are A082402.

%Y Leading diagonal is A097629.

%Y The unlabeled version is A350449.

%Y Cf. A057273, A062735, A081064.

%K nonn,tabf

%O 1,3

%A _Andrew Howroyd_, Jan 29 2022