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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.
14
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
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
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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Feb 23 2019
STATUS
approved