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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

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