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).

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).

Links

Table of n, a(n) for n=1..84.

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

Adjacent sequences: A214848 A214849 A214850 * A214852 A214853 A214854

Keyword

nonn,tabf

Author

Geoffrey Critzer, Mar 08 2013

Status

approved