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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).
2
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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
OFFSET
1,6
COMMENTS
Row sums = n!.
Sum_{k=1...A000085(n)} T(n,k)*k = n!.
Sum_{k=1...A000085(n)} T(n,k) = A003483(n).
Column k=0 is n! - A003483(n).
EXAMPLE
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
450, 184, 0, 0, 85, 0,0,0,...,1 where the 1 is in column k=76.
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.
MATHEMATICA
(* 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
CROSSREFS
Cf. A214849 (column k=1), A214854 (column k=2).
Sequence in context: A292380 A242165 A231724 * A245203 A133949 A139808
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Mar 08 2013
STATUS
approved