A003084 Related to number of digraphs.

1, 5, 40, 801, 46821, 9185102, 6163297995, 14339791643249, 117235455142196308, 3412474003994007703605, 357748249084029269153547905, 136400554886800212073525651823742, 190697966236731843091458826668123014367, 984418987245772021436902193577676975221669509, 18875177868521443706244256784212908480749407027875180

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 124, table 5.1.2, p*a_p
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Programs

Mathematica

Needs["Combinatorica`"]; d[n_] := GraphPolynomial[n, x, Directed] /. x -> 1; max = 12; se = Series[ Sum[ a[n]*x^n/n, {n, 1, max}] - Log[1 + Sum[ d[n]*x^n, {n, 1, max}]], {x, 0, max}]; sol = SolveAlways[ se == 0, x]; A003084 = Table[ a[n], {n, 1, max}] /. sol[[1]] (* Jean-François Alcover, Feb 01 2012, after formula *)
terms = 15;
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[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
d[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]} ]; s/n!);
CoefficientList[Log[Sum[ d[n] x^n, {n, 0, terms + 1}]] + O[x]^(terms + 1), x] Range[0, terms] // Rest (* Jean-François Alcover, Aug 29 2019, after Andrew Howroyd in A000273 *)

Formula

Sum a(n) x^n / n = log (1 + Sum d(n) x^n ), where d(n) is # digraphs on n nodes (A000273).

Extensions

Corrected and extended by Vladeta Jovovic, Jan 09 2000

More terms from Jean-François Alcover, Aug 29 2019