A324362 Total number of occurrences of k in the (signed) displacement sets of all permutations of [n+k] divided by k!; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 3, 4, 0, 1, 5, 13, 15, 0, 1, 7, 28, 67, 76, 0, 1, 9, 49, 179, 411, 455, 0, 1, 11, 76, 375, 1306, 2921, 3186, 0, 1, 13, 109, 679, 3181, 10757, 23633, 25487, 0, 1, 15, 148, 1115, 6576, 29843, 98932, 214551, 229384, 0, 1, 17, 193, 1707, 12151, 69299, 307833, 1006007, 2160343, 2293839

Offset

0,9

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Wikipedia, Permutation

Formula

E.g.f. of column k: (1-exp(-x))/(1-x)^(k+1).

A(n,k) = -1/k! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+k-j)!.

A(n,k) = A306234(n+k,k).

Example

Square array A(n,k) begins:
    0,    0,     0,     0,     0,      0,      0, ...
    1,    1,     1,     1,     1,      1,      1, ...
    1,    3,     5,     7,     9,     11,     13, ...
    4,   13,    28,    49,    76,    109,    148, ...
   15,   67,   179,   375,   679,   1115,   1707, ...
   76,  411,  1306,  3181,  6576,  12151,  20686, ...
  455, 2921, 10757, 29843, 69299, 142205, 266321, ...

Maple

A:= (n, k)-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!:
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

m = 10;
col[k_] := col[k] = CoefficientList[(1-Exp[-x])/(1-x)^(k+1)+O[x]^(m+1), x]* Range[0, m]!;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 03 2021 *)

Crossrefs

Columns k=0-10 give: A002467, A180191(n+1), A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.

Rows n=0-3 give: A000004, A000012, A005408, A056107(k+1).

Main diagonal gives A324361.

Cf. A306234.

Sequence in context: A170952 A194587 A175646 * A073234 A123685 A124917

Adjacent sequences: A324359 A324360 A324361 * A324363 A324364 A324365

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 23 2019

Status

approved