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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.
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%I #24 May 03 2021 10:10:42

%S 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,

%T 0,1,11,76,375,1306,2921,3186,0,1,13,109,679,3181,10757,23633,25487,0,

%U 1,15,148,1115,6576,29843,98932,214551,229384,0,1,17,193,1707,12151,69299,307833,1006007,2160343,2293839

%N 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.

%H Alois P. Heinz, <a href="/A324362/b324362.txt">Antidiagonals n = 0..140, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>

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

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

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

%e Square array A(n,k) begins:

%e 0, 0, 0, 0, 0, 0, 0, ...

%e 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 3, 5, 7, 9, 11, 13, ...

%e 4, 13, 28, 49, 76, 109, 148, ...

%e 15, 67, 179, 375, 679, 1115, 1707, ...

%e 76, 411, 1306, 3181, 6576, 12151, 20686, ...

%e 455, 2921, 10757, 29843, 69299, 142205, 266321, ...

%p A:= (n, k)-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!:

%p seq(seq(A(n, d-n), n=0..d), d=0..12);

%t m = 10;

%t col[k_] := col[k] = CoefficientList[(1-Exp[-x])/(1-x)^(k+1)+O[x]^(m+1), x]* Range[0, m]!;

%t A[n_, k_] := col[k][[n+1]];

%t Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 03 2021 *)

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

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

%Y Main diagonal gives A324361.

%Y Cf. A306234.

%K nonn,tabl

%O 0,9

%A _Alois P. Heinz_, Feb 23 2019