%I #26 Oct 28 2022 14:22:59
%S 2,2,4,6,8,8,22,28,24,16,90,112,96,64,32,394,484,416,288,160,64,1806,
%T 2200,1896,1344,800,384,128,8558,10364,8952,6448,4000,2112,896,256,
%U 41586,50144,43392,31616,20160,11264,5376,2048,512
%N Triangle read by rows: T(n,k) = number of Schroeder (or royal) n-paths (A006318) containing k returns to the diagonal y=x. (A northeast step lying on y=x contributes a return.)
%H Michael De Vlieger, <a href="/A108891/b108891.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)
%H Scott Balchin, Ethan MacBrough, and Kyle Ormsby, <a href="https://arxiv.org/abs/2209.06992">The combinatorics of N_oo operads for C_qp^n and D_p^n</a>, arXiv:2209.06992 [math.AT], 2022.
%F Column k is the k-fold convolution of column 1.
%F T(n, k) = A104219(n-1, k-1)*2^k. - _Philippe Deléham_, Jul 31 2005
%F Triangle T(n,k), 1 <= k <= n, read by rows given by (0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 02 2013
%e Table begins
%e n\k 1 2 3 4 5 6
%e -------------------------------
%e 1 | 2
%e 2 | 2 4
%e 3 | 6 8 8
%e 4 | 22 28 24 16
%e 5 | 90 112 96 64 32
%e 6 |394 484 416 288 160 64
%e The paths DD, END, DEN, ENEN each have 2 returns (E=east, N=north, D=northeast); so T(2,2)=4.
%e From _Philippe Deléham_, Nov 02 2013: (Start)
%e Triangle (0, 1, 2, 1, 2, 1, 2, ...) DELTA (1, 0, 0, 0, ...) begins:
%e 1;
%e 0, 2;
%e 0, 2, 4;
%e 0, 6, 8, 8;
%e 0, 22, 28, 24, 16;
%e 0, 90, 112, 96, 64, 32;
%e 0, 394, 484, 416, 288, 160, 64; (End)
%t T[n_, k_] := (-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 2]; Table[T[n - 1, k - 1]*2^k, {n, 9}, {k, n}] // Flatten (* _Michael De Vlieger_, Sep 21 2022, after _Peter Luschny_ at A104219 *)
%Y Row sums are the large Schroeder numbers A006318. Column k=1 is twice the little Schroeder numbers A001003. The main diagonal consists of powers of 2, A000079. The first subdiagonal is A036289. The analogous Catalan triangle is A009766 (with rows reversed).
%K nonn,tabl
%O 1,1
%A _David Callan_, Jul 25 2005