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Number of 2-colored labeled directed acyclic graphs on n nodes such that all black nodes are sources.
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%I #39 Oct 08 2022 22:17:01

%S 1,2,8,74,1664,90722,11756288,3544044674,2439773425664,

%T 3777981938265602,12999312305021800448,98399334883456516073474,

%U 1625096032161083727093530624,58150966795467956854830216929282

%N Number of 2-colored labeled directed acyclic graphs on n nodes such that all black nodes are sources.

%C A source is a vertex with in-degree equal to 0. There may be white sources as well.

%H Valery A. Liskovets, <a href="https://arxiv.org/abs/0804.2496">More on counting acyclic digraphs</a>, arXiv:0804.2496 [math.CO], 2008.

%F a(n) = sum_{t=0..n}binomial(n,t)*2^(t(n-t))*A003024(t).

%F Let E(x) = sum_{n>=0}x^n/(2^n*n!). Then sum_{n>=0}a(n)x^n/(2^n*n!) = E(x)/E(-x).

%F a(n) = sum_{k=0..n}|A224069(n,k)|.

%t nn = 13; B[n_, q_] := q^Binomial[n, 2] n!; e[u_, q_] := Sum[u^n/B[n, q], {n, 0, nn}];Table[B[n, 2], {n, 0, nn}] CoefficientList[Series[e[u, 2]/e[-u, 2], {u, 0, nn}], u]

%Y Cf. A003024

%K nonn

%O 0,2

%A _Geoffrey Critzer_, Oct 08 2022