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Number of ordered triples (a,b,c) of elements of the symmetric group S_n such that the triple a,b,c generates a transitive group.
2

%I #23 Jul 11 2020 07:39:32

%S 1,7,194,12858,1647384,361351560,125116670160,64439768489040,

%T 47159227114392960,47285264408385951360,63057420721939066617600,

%U 109118766834521171299756800,239996135160204867851157273600,659114500480471292127627441484800

%N Number of ordered triples (a,b,c) of elements of the symmetric group S_n such that the triple a,b,c generates a transitive group.

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; page 139.

%F E.g.f.: log(Sum_{n>=0} n!^2*x^n).

%F a(n) = (n!)^3 - (n-1)! * Sum_{k=1..n-1} a(k) * ((n-k)!)^2 / (k-1)!. - _Ilya Gutkovskiy_, Jul 10 2020

%t nn=14; b=Sum[n!^3 x^n/n!,{n,0,nn}]; Drop[Range[0,nn]!CoefficientList[Series[Log[b],{x,0,nn}],x],1]

%o (PARI)

%o N = 66; x = 'x + O('x^N);

%o egf = log(sum(n=0, N, n!^2*x^n));

%o gf = serlaplace(egf);

%o v = Vec(gf)

%o /* _Joerg Arndt_, Apr 14 2013 */

%Y Cf. A122949, A071605.

%K nonn

%O 1,2

%A _Geoffrey Critzer_, Apr 13 2013