This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001174 M1809 N0715 #55 Feb 16 2025 08:32:22
%S A001174 1,2,7,42,582,21480,2142288,575016219,415939243032,816007449011040,
%T A001174 4374406209970747314,64539836938720749739356,
%U A001174 2637796735571225009053373136,300365896158980530053498490893399
%N A001174 Number of oriented graphs (i.e., digraphs with no bidirected edges) on n unlabeled nodes. Also number of complete digraphs on n unlabeled nodes. Number of antisymmetric relations (i.e., oriented graphs with loops) on n unlabeled nodes is A083670.
%D A001174 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 133, c_p.
%D A001174 M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
%D A001174 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001174 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001174 Andrew Howroyd, <a href="/A001174/b001174.txt">Table of n, a(n) for n = 1..50</a>
%H A001174 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A001174 R. L. Davis, <a href="http://dx.doi.org/10.1090/S0002-9939-1953-0055294-2">The number of structures of finite relations</a>, Proc. Amer. Math. Soc. 4 (1953), 486-495.
%H A001174 Musa Demirci, Ugur Ana, and Ismail Naci Cangul, <a href="https://doi.org/10.1007/978-981-16-1402-6">Properties of Characteristic Polynomials of Oriented Graphs</a>, Proc. Int'l Conf. Adv. Math. Comp. (ICAMC 2020) Springer, see p. 60.
%H A001174 F. Harary and E. M. Palmer, <a href="http://dx.doi.org/10.1090/S0002-9939-1966-0191845-4">Enumeration of mixed graphs</a>, Proc. Amer. Math. Soc., 17 (1966), 682-687.
%H A001174 T. R. Hoffman and J. P. Solazzo, <a href="http://arxiv.org/abs/1408.0334">Complex Two-Graphs via Equiangular Tight Frames</a>, arXiv preprint arXiv:1408.0334 [math.CO], 2014-2017.
%H A001174 M. D. McIlroy, <a href="/A000088/a000088.pdf">Calculation of numbers of structures of relations on finite sets</a>, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
%H A001174 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A001174 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OrientedGraph.html">Oriented Graph</a>
%F A001174 There's an explicit formula - see for example Harary and Palmer (book), Eq. (5.4.14).
%F A001174 a(n) ~ 3^(n*(n-1)/2)/n! [McIlroy, 1955]. - _Vaclav Kotesovec_, Dec 19 2016
%t A001174 permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t A001174 edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v - 1, 2];
%t A001174 a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!];
%t A001174 Array[a, 15] (* _Jean-François Alcover_, Jul 06 2018, after _Andrew Howroyd_ *)
%o A001174 (PARI)
%o A001174 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 A001174 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 A001174 a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)); s/n!} \\ _Andrew Howroyd_, Oct 23 2017
%o A001174 (Python)
%o A001174 from itertools import combinations
%o A001174 from math import prod, gcd, factorial
%o A001174 from fractions import Fraction
%o A001174 from sympy.utilities.iterables import partitions
%o A001174 def A001174(n): return int(sum(Fraction(3**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q-1>>1)*r+(q*r*(r-1)>>1) for q, r in p.items())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # _Chai Wah Wu_, Jul 15 2024
%Y A001174 Cf. A000595, A001173, A281446.
%Y A001174 Cf. A047656 (labeled case), A054941 (connected labeled case), A086345 (connected unlabeled case).
%K A001174 nonn,nice,easy
%O A001174 1,2
%A A001174 _N. J. A. Sloane_
%E A001174 More terms from _Vladeta Jovovic_
%E A001174 Revised description from _Vladeta Jovovic_, Jan 20 2005