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A122949 Number of ordered pairs of permutations generating a transitive group.

%I A122949 #27 Jul 11 2020 07:39:24
%S A122949 1,3,26,426,11064,413640,20946960,1377648720,114078384000,
%T A122949 11611761920640,1425189271161600,207609729886944000,
%U A122949 35419018603306060800,6996657393055480550400,1584616114318716544665600,407930516160959891683584000,118458533875304716189544448000
%N A122949 Number of ordered pairs of permutations generating a transitive group.
%C A122949 From Dixon: The sequence is asymptotic to (n!)^2; when divided by n!^2, it has a high-order asymptotic contact with the probability that two randomly chosen permutations generate the symmetric group. Also: a(n)=(n-1)!*A003319(n+1), where A003319 is the number of connected [or indecomposable] permutations. The coefficients in the asymptotic expansion of a(n)/(n!)^2 are A113869 and in absolute value, they constitute A084357 (number of sets of sets of lists).
%H A122949 Seiichi Manyama, <a href="/A122949/b122949.txt">Table of n, a(n) for n = 1..253</a>
%H A122949 John D. Dixon, <a href="http://www.combinatorics.org/Volume_12/Abstracts/v12i1r56.html">Asymptotics of Generating the Symmetric and Alternating Groups</a>, Electronic Journal of Combinatorics, vol 11(2), R56.
%H A122949 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; page 139.
%F A122949 Exponential generating function is: log(1+Sum_{n>=1}n!*z^n).
%F A122949 a(n) = (n!)^2 - (n-1)! * Sum_{k=1..n-1} a(k) * (n-k)! / (k-1)!. - _Ilya Gutkovskiy_, Jul 10 2020
%e A122949 a(2)=3 because there are 2!*2!=4 pairs of permutations, of which only [(1,1),(1,1)] does not generate a transitive group.
%p A122949 series(log(add(n!*z^n,n=0..Order+2)),z=0):seq(coeff(%,z,j)*j!,j=0..Order);
%t A122949 max = 15; Drop[ CoefficientList[ Series[ Log[1 + Sum[n!*z^n, {n, 1, max}]], {z, 0, max}], z]* Range[0, max]!, 1](* _Jean-François Alcover_, Oct 05 2011 *)
%o A122949 (PARI) N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, k!*x^k)))) \\ _Seiichi Manyama_, Mar 01 2019
%Y A122949 Cf. A003319, A084357, A113869.
%K A122949 nonn
%O A122949 1,2
%A A122949 _Philippe Flajolet_, Oct 25 2006
%E A122949 More terms from _Seiichi Manyama_, Mar 01 2019