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A101389 Number of n-vertex unlabeled oriented graphs without endpoints.

%I A101389 #14 Jan 25 2021 19:56:56
%S A101389 1,1,1,3,21,369,16929,1913682,546626268,406959998851,808598348346150,
%T A101389 4358157210587930509,64443771774627635711718,
%U A101389 2636248889492487709302815665,300297332862557660078111708007894,95764277032243987785712142452776403618,85885545190811866954428990373255822969983915
%N A101389 Number of n-vertex unlabeled oriented graphs without endpoints.
%H A101389 Andrew Howroyd, <a href="/A101389/b101389.txt">Table of n, a(n) for n = 0..50</a>
%e A101389 a(3) = 3 because there are 2 distinct orientations of the triangle K_3 plus the empty graph on 3 vertices.
%o A101389 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A101389 oedges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
%o A101389 ographsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 3^oedges(p) * sMonomial(p)); s/n!}
%o A101389 ographs(n)={sum(k=0, n, ographsCycleIndex(k)*x^k) + O(x*x^n)}
%o A101389 trees(n,k)={sRevert(x*sv(1)/sExp(k*x*sv(1) + O(x^n)))}
%o A101389 cycleIndexSeries(n)={my(g=ographs(n), tr=trees(n,2), tu=tr-tr^2); sSolve( g/sExp(tu), tr )*symGroupSeries(n)}
%o A101389 NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ _Andrew Howroyd_, Dec 27 2020
%o A101389 (PARI) \\ faster stand-alone version
%o A101389 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A101389 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
%o A101389 seq(n)={Vec(sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 3^edges(p) * prod(i=1, #p, my(d=p[i]); (1-x^d)^2 + O(x*x^(n-k))) ); x^k*s/k!)/(1-x^2))} \\ _Andrew Howroyd_, Jan 22 2021
%Y A101389 Cf. A100569 (labeled case), A100548, A101388, A004110, A004108, A059166, A059167, A001174.
%K A101389 nonn
%O A101389 0,4
%A A101389 Goran Kilibarda, _Vladeta Jovovic_, Jan 14 2005
%E A101389 a(0)=1 prepended and terms a(9) and beyond from _Andrew Howroyd_, Dec 27 2020