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A137435 Acyclic 3-multidigraphs on n nodes.

%I A137435 #37 Mar 21 2019 10:13:26
%S A137435 1,1,7,289,63487,69711361,367404658687,9036285693861889,
%T A137435 1015983915928423497727,514039127264534042076119041,
%U A137435 1155907276780291114251550828003327,11436746463485293365165228859824053157889,493776641438913029616304251647570171691844763647
%N A137435 Acyclic 3-multidigraphs on n nodes.
%C A137435 This is the 2nd row of Table 1, p. 3 in Liskovets. The first row is A003024.
%H A137435 Vincenzo Librandi, <a href="/A137435/b137435.txt">Table of n, a(n) for n = 0..55</a>
%H A137435 Valery A. Liskovets, <a href="https://arxiv.org/abs/0804.2496">More on counting acyclic digraphs</a>, arXiv:0804.2496 [math.CO], 2008.
%F A137435 1 = Sum_{n>=0} a(n)*exp(-4^n*x)*x^n/n!. - _Vladeta Jovovic_, Apr 22 2008
%F A137435 1 = Sum_{n>=0} a(n)*x^n/(1 + 4^n*x)^(n+1). - _Paul D. Hanna_, Oct 17 2009
%F A137435 1 = Sum_{n>=0} a(n)*binomial(n+m-1,n)*x^n/(1 + 4^n*x)^(n+m) for m >= 1. - _Paul D. Hanna_, Apr 01 2011
%F A137435 log(1+x) = Sum_{n>=1} a(n)*(x^n/n)/(1 + 4^n*x)^n. - _Paul D. Hanna_, Apr 01 2011
%F A137435 a(n) = Sum_{k=1..n} (-1)^(k+1)*C(n, k)*4^(k*(n-k))*a(n-k) for n > 0 with a(0)=1. - _Paul D. Hanna_, Apr 01 2011
%e A137435 From _Paul D. Hanna_, Apr 01 2011: (Start)
%e A137435 Illustration of the generating functions.
%e A137435 E.g.f.: 1 = exp(-x) + exp(-4*x)*x + 7*exp(-16*x)*x^2/2! + 289*exp(-64*x)*x^3/3! + ...
%e A137435 L.g.f.: log(1+x) = x/(1+4*x) + 7*(x^2/2)/(1+16*x)^2 + 289*(x^3/3)/(1+64*x)^3 + ...
%e A137435 G.f.: 1 = 1/(1+x) + 1*x/(1+4*x)^2 + 7*x^2/(1+16*x)^3 + 289*x^3/(1+64*x)^4 + ...
%e A137435 G.f.: 1 = 1/(1+x)^2 + 1*2*x/(1+4*x)^3 + 7*3*x^2/(1+16*x)^4 + 289*4*x^3/(1+64*x)^5 + ...
%e A137435 G.f.: 1 = 1/(1+x)^3 + 1*3*x/(1+4*x)^4 + 7*6*x^2/(1+16*x)^5 + 289*10*x^3/(1+64*x)^6 + ... (End)
%t A137435 a[n_] := a[n] = Sum[(-1)^(k+1)*Binomial[n, k]*4^(k*(n-k))*a[n-k], {k, 1, n}]; a[0]=1; Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Dec 15 2014, after _Paul D. Hanna_ *)
%o A137435 (PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+4^k*x+x*O(x^n))^(k+1)), n)} \\ _Paul D. Hanna_, Oct 17 2009
%o A137435 (PARI) /* Holds for m>=1: */
%o A137435 {a(n)=local(m=1); polcoeff(1-sum(k=0, n-1, a(k)*binomial(m+k-1, k)*x^k/(1+4^k*x+x*O(x^n))^(k+m)), n)/binomial(m+n-1, n)} \\ _Paul D. Hanna_, Apr 01 2011
%o A137435 (PARI) /* Recurrence: */
%o A137435 {a(n)=if(n<1, n==0, sum(k=1, n, -(-1)^k*binomial(n, k)*4^(k*(n-k))*a(n-k)))} \\ _Paul D. Hanna_, Apr 01 2011
%o A137435 (PARI) /* E.g.f.: */
%o A137435 {a(n)=n!*polcoeff(1-sum(k=0, n-1, a(k)*exp(-4^k*x+x*O(x^n))*x^k/k!), n)} \\ _Paul D. Hanna_, Apr 01 2011
%Y A137435 Cf. A003024, A188457.
%K A137435 nonn
%O A137435 0,3
%A A137435 _Jonathan Vos Post_, Apr 17 2008
%E A137435 More terms from _Vladeta Jovovic_, Apr 22 2008
%E A137435 Offset changed to 0 by _Paul D. Hanna_, Apr 01 2011