%I #20 Jan 03 2021 16:55:56
%S 1,0,1,0,3,1,0,26,9,1,0,426,131,18,1,0,11064,2910,395,30,1,0,413640,
%T 92314,11475,925,45,1,0,20946960,3980172,438424,34125,1855,63,1,0,
%U 1377648720,224782284,21632436,1550689,84840,3346,84,1,0,114078384000,16158371184,1353378284,87036012,4533249,185976,5586,108,1
%N Triangular array read by rows: T(n,k) is the number of ordered pairs of n-permutations that generate a group with exactly k orbits, 0 <= k <= n, n >= 0.
%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.: exp(y*log(Sum_{n>=0} n! * x^n)).
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 3, 1;
%e 0, 26, 9, 1;
%e 0, 426, 131, 18, 1;
%e 0, 11064, 2910, 395, 30, 1;
%e 0, 413640, 92314, 11475, 925, 45, 1;
%e T(3,2) = 9 because we have 3 ordered pairs (e,<(1,2)>), (<(1,2)>,e), (<(1,2)>,<(1,2)>) for each of the 3 transpositions in S_3.
%t nn = 7; Range[0, nn]! CoefficientList[Series[Exp[u Log[Sum[n!^2 z^n/n!, {n, 0, nn}]]], {z, 0, nn}], {z, u}] // Grid
%Y Cf. A122949 (column 1), A001044 (row sums), A220754.
%K nonn,tabl
%O 0,5
%A _Geoffrey Critzer_, Sep 16 2019