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A003084 Related to number of digraphs.

%I A003084 M3993 #25 Dec 26 2021 20:39:59
%S A003084 1,5,40,801,46821,9185102,6163297995,14339791643249,
%T A003084 117235455142196308,3412474003994007703605,
%U A003084 357748249084029269153547905,136400554886800212073525651823742,190697966236731843091458826668123014367,984418987245772021436902193577676975221669509,18875177868521443706244256784212908480749407027875180
%N A003084 Related to number of digraphs.
%D A003084 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 124, table 5.1.2, p*a_p
%D A003084 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%F A003084 Sum a(n) x^n / n = log (1 + Sum d(n) x^n ), where d(n) is # digraphs on n nodes (A000273).
%t A003084 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 *)
%t A003084 terms = 15;
%t A003084 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 A003084 edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
%t A003084 d[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]} ]; s/n!);
%t A003084 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 *)
%K A003084 nonn
%O A003084 1,2
%A A003084 _N. J. A. Sloane_
%E A003084 Corrected and extended by _Vladeta Jovovic_, Jan 09 2000
%E A003084 More terms from _Jean-François Alcover_, Aug 29 2019