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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.

%I A324362 #24 May 03 2021 10:10:42
%S A324362 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 A324362 0,1,11,76,375,1306,2921,3186,0,1,13,109,679,3181,10757,23633,25487,0,
%U A324362 1,15,148,1115,6576,29843,98932,214551,229384,0,1,17,193,1707,12151,69299,307833,1006007,2160343,2293839
%N 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.
%H A324362 Alois P. Heinz, <a href="/A324362/b324362.txt">Antidiagonals n = 0..140, flattened</a>
%H A324362 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A324362 E.g.f. of column k: (1-exp(-x))/(1-x)^(k+1).
%F A324362 A(n,k) = -1/k! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+k-j)!.
%F A324362 A(n,k) = A306234(n+k,k).
%e A324362 Square array A(n,k) begins:
%e A324362     0,    0,     0,     0,     0,      0,      0, ...
%e A324362     1,    1,     1,     1,     1,      1,      1, ...
%e A324362     1,    3,     5,     7,     9,     11,     13, ...
%e A324362     4,   13,    28,    49,    76,    109,    148, ...
%e A324362    15,   67,   179,   375,   679,   1115,   1707, ...
%e A324362    76,  411,  1306,  3181,  6576,  12151,  20686, ...
%e A324362   455, 2921, 10757, 29843, 69299, 142205, 266321, ...
%p A324362 A:= (n, k)-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!:
%p A324362 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A324362 m = 10;
%t A324362 col[k_] := col[k] = CoefficientList[(1-Exp[-x])/(1-x)^(k+1)+O[x]^(m+1), x]* Range[0, m]!;
%t A324362 A[n_, k_] := col[k][[n+1]];
%t A324362 Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 03 2021 *)
%Y A324362 Columns k=0-10 give: A002467, A180191(n+1), A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.
%Y A324362 Rows n=0-3 give: A000004, A000012, A005408, A056107(k+1).
%Y A324362 Main diagonal gives A324361.
%Y A324362 Cf. A306234.
%K A324362 nonn,tabl
%O A324362 0,9
%A A324362 _Alois P. Heinz_, Feb 23 2019