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A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows. 4
1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 18, 24, 18, 6, 1, 1, 8, 36, 96, 120, 96, 36, 8, 1, 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1, 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 35280, 15120, 4200, 840, 126, 14, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Alois P. Heinz, Rows n = 1..100, flattened

Wikipedia, Permutation

Wikipedia, Permutation matrix

FORMULA

T(n,k) = T(n,-k).

T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.

T(n,k) = 1/|k|! * A324225(n,k).

E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.

Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).

EXAMPLE

Triangle T(n,k) begins:

: 1 ;

: 1, 2, 1 ;

: 1, 4, 6, 4, 1 ;

: 1, 6, 18, 24, 18, 6, 1 ;

: 1, 8, 36, 96, 120, 96, 36, 8, 1 ;

: 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1 ;

: 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1 ;

MAPLE

b:= proc(s, c) option remember; (n-> `if`(n=0, c,

add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))

end:

T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):

seq(T(n), n=1..8);

# second Maple program:

egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):

T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):

seq(seq(T(n, k), k=1-n..n-1), n=1..8);

# third Maple program:

T:= (n, k)-> (t-> `if`(t<n, (n-t)*(n-1)!/t!, 0))(abs(k)):

seq(seq(T(n, k), k=1-n..n-1), n=1..8);

MATHEMATICA

T[n_, k_] := With[{t = Abs[k]}, If[t<n, (n-t)(n-1)!/t!, 0]];

Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 25 2021, after 3rd Maple program *)

CROSSREFS

Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.

Row sums give A306495(n-1).

Cf. A306234, A324225.

Sequence in context: A327639 A273891 A034870 * A264622 A275017 A141036

Adjacent sequences: A324221 A324222 A324223 * A324225 A324226 A324227

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Feb 18 2019

STATUS

approved

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Last modified February 8 11:37 EST 2023. Contains 360138 sequences. (Running on oeis4.)