%I #30 Apr 17 2022 03:55:29
%S 1,1,0,1,1,0,1,2,3,0,1,3,10,19,0,1,4,21,92,211,0,1,5,36,255,1354,3651,
%T 0,1,6,55,544,4725,29252,90921,0,1,7,78,995,12196,123903,873964,
%U 3081513,0,1,8,105,1644,26215,377904,4368729,34555880,136407699,0
%N Square array read by descending antidiagonals. T(n,k) is the number of ways to separate the columns of an ordered pair of n-permutations (that have been written as a 2 X n array, one atop the other) into k cells so that no cell has a column rise. For n >= 0, k >= 0.
%C A column rise (cf. A000275) means a pair of adjacent columns within a cell where each entry in the first column is less than the adjacent entry in the second column. The order of the columns cannot change. The cells are allowed to be empty.
%D R. P. Stanley, Enumerative Combinatorics, Vol. I, Second Edition, Section 3.13.
%H Alois P. Heinz, <a href="/A340986/b340986.txt">Antidiagonals n = 0..100, flattened</a>
%F Let E(x) = Sum_{n>=0} x^n/n!^2. Then Sum_{n>=0} T(n,k)*x^n/n!^2 = 1/E(-x)^k.
%F T(n,k) = (n!)^2 * [x^n] 1/BesselJ(0,2*sqrt(x))^k. - _Alois P. Heinz_, Feb 02 2021
%e Square array T(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, ...
%e 0, 3, 10, 21, 36, 55, ...
%e 0, 19, 92, 255, 544, 995, ...
%e 0, 211, 1354, 4725, 12196, 26215, ...
%e 0, 3651, 29252, 123903, 377904, 939155, ...
%p T:= (n, k)-> n!^2*coeff(series(1/BesselJ(0, 2*sqrt(x))^k, x, n+1), x, n):
%p seq(seq(T(n, d-n), n=0..d), d=0..10); # _Alois P. Heinz_, Feb 02 2021
%t nn = 6; B[n_] := n!^2; e[x_] := Sum[x^n/B[n], {n, 0, nn}];
%t Table[Table[B[n], {n, 0, nn}] PadRight[CoefficientList[Series[e[-x]^-k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Grid
%Y Columns k=0-4 give: A000007, A000275, A336271, A336638, A336639.
%Y Rows n=0-2 give: A000012, A001477, A014105.
%Y Main diagonal gives A336665.
%Y Cf. A212855, A192721, A334394.
%K nonn,tabl
%O 0,8
%A _Geoffrey Critzer_, Feb 01 2021