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A214851 Irregular triangular array read by rows. T(n,k) is the number of n-permutations that have exactly k square roots. n >= 1, 0 <= k <= A000085(n).

%I A214851 #17 Apr 12 2013 12:34:48
%S A214851 0,1,1,0,1,3,2,0,0,1,12,8,3,0,0,0,0,0,0,0,1,60,24,35,0,0,0,0,0,0,0,0,
%T A214851 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,450,184,0,0,85,0,0,0,0,0,0,0,0,0,0,0,
%U A214851 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A214851 Irregular triangular array read by rows.  T(n,k) is the number of n-permutations that have exactly k square roots.  n >= 1, 0 <= k <= A000085(n).
%C A214851 Row sums = n!.
%C A214851 Sum_{k=1...A000085(n)} T(n,k)*k = n!.
%C A214851 Sum_{k=1...A000085(n)} T(n,k) = A003483(n).
%C A214851 Column k=0 is n! - A003483(n).
%e A214851 0, 1,
%e A214851 1, 0, 1,
%e A214851 3, 2, 0, 0, 1,
%e A214851 12, 8, 3, 0, 0, 0, 0, 0, 0, 0, 1,
%e A214851 60, 24, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
%e A214851 450, 184, 0, 0, 85, 0,0,0,...,1 where the 1 is in column k=76.
%e A214851 T(5,2)= 35 because we have 20 5-permutations of the type (1,2,3)(4)(5) and 15 of the type (1,2)(3,4)(5).  These have 2 square roots:(1,3,2)(4)(5),(1,3,2)(4,5) and (1,3,2,4)(5),(3,1,4,2)(5) respectively.
%t A214851 (* Warning: the code is very inefficient, it takes about one minute to run on a laptop computer. *) a={1,2,4,10,26}; Table[Distribution[Distribution[Table[MultiplicationTable[Permutations[m], Permute[#1,#2]&][[n]][[n]], {n,1,m!}], Range[1,m!]], Range[0,a[[m]]]], {m,1,5}] //Grid
%Y A214851 Cf. A214849 (column k=1), A214854 (column k=2).
%K A214851 nonn,tabf
%O A214851 1,6
%A A214851 _Geoffrey Critzer_, Mar 08 2013