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A350487 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n labeled nodes with k arcs and a global source, n >= 1, k = 0..n*(n-1)/2.

%I A350487 #12 Jan 08 2022 19:24:37
%S A350487 1,0,2,0,0,9,6,0,0,0,64,132,96,24,0,0,0,0,625,2640,4850,4900,2850,900,
%T A350487 120,0,0,0,0,0,7776,55800,186480,379170,516660,491040,328680,152640,
%U A350487 46980,8640,720,0,0,0,0,0,0,117649,1286670,6756120,22466010
%N A350487 Triangle read by rows: T(n,k) is the number of acyclic digraphs on n labeled nodes with k arcs and a global source, n >= 1, k = 0..n*(n-1)/2.
%H A350487 Andrew Howroyd, <a href="/A350487/b350487.txt">Table of n, a(n) for n = 1..1350</a> (rows 1..20)
%H A350487 Marcel et al., <a href="https://mathoverflow.net/q/395095">Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?</a>, MathOverflow, 2021.
%e A350487 Triangle begins:
%e A350487   [1] 1;
%e A350487   [2] 0, 2;
%e A350487   [3] 0, 0, 9,  6;
%e A350487   [4] 0, 0, 0, 64, 132,   96,   24;
%e A350487   [5] 0, 0, 0,  0, 625, 2640, 4850, 4900, 2850, 900, 120;
%e A350487   ...
%o A350487 (PARI)
%o A350487 T(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1)*binomial(n,k)*((1+'y)^(n-k)-1)^k*a[n-k])); [Vecrev(p) | p <- a]}
%o A350487 { my(A=T(6)); for(n=1, #A, print(A[n])) }
%Y A350487 Row sums are A003025.
%Y A350487 Leading diagonal is A000169.
%Y A350487 The unlabeled version is A350488.
%Y A350487 Cf. A081064.
%K A350487 nonn,tabf
%O A350487 1,3
%A A350487 _Andrew Howroyd_, Jan 01 2022