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 A086345 Number of connected oriented graphs (i.e., connected directed graphs with no bidirected edges) on n nodes. 9
 1, 1, 1, 5, 34, 535, 20848, 2120098, 572849763, 415361983540, 815590925440865, 4373589784210012634, 64535461714821630421106, 2637732191356603658136444467, 300363258297687600380548275359231 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..50 Musa Demirci, Ugur Ana, and Ismail Naci Cangul, Properties of Characteristic Polynomials of Oriented Graphs, Proc. Int'l Conf. Adv. Math. Comp. (ICAMC 2020) Springer, see p. 61. Chathura Kankanamge, Multiple Continuous Subgraph Query Optimization Using Delta Subgraph Queries, Master Maths Thesis, Univ. of Waterloo, 2018. Eric Weisstein's World of Mathematics, Oriented Graph FORMULA Inverse Euler transform of A001174. - Andrew Howroyd, Nov 03 2017 MATHEMATICA 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]; edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total @ Quotient[v - 1, 2]; a1174[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p], {p, IntegerPartitions[n]}]; s/n!]; b = Array[a1174, 15]; mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0]; EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]]; A086345 = EULERi[b] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *) PROG (Python) from functools import lru_cache from itertools import combinations from math import prod, factorial, gcd from fractions import Fraction from sympy.utilities.iterables import partitions from sympy import mobius, divisors def A086345(n): @lru_cache(maxsize=None) def b(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))) @lru_cache(maxsize=None) def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1, n)) return sum(mobius(d)*c(n//d) for d in divisors(n, generator=True))//n if n else 1 # Chai Wah Wu, Jul 15 2024 CROSSREFS Cf. A054941 (labeled case), A001174, A281446 (multisets). Sequence in context: A211037 A130607 A211042 * A295545 A309534 A348375 Adjacent sequences: A086342 A086343 A086344 * A086346 A086347 A086348 KEYWORD nonn AUTHOR Eric W. Weisstein, Jul 16 2003 EXTENSIONS More terms from Vladeta Jovovic, Jul 19 2003 a(0)=1 prepended by Andrew Howroyd, Sep 09 2018 STATUS approved

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Last modified August 3 05:44 EDT 2024. Contains 374875 sequences. (Running on oeis4.)