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A108891 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.)

%I A108891 #26 Oct 28 2022 14:22:59
%S A108891 2,2,4,6,8,8,22,28,24,16,90,112,96,64,32,394,484,416,288,160,64,1806,
%T A108891 2200,1896,1344,800,384,128,8558,10364,8952,6448,4000,2112,896,256,
%U A108891 41586,50144,43392,31616,20160,11264,5376,2048,512
%N A108891 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 A108891 Michael De Vlieger, <a href="/A108891/b108891.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)
%H A108891 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 A108891 Column k is the k-fold convolution of column 1.
%F A108891 T(n, k) = A104219(n-1, k-1)*2^k. - _Philippe Deléham_, Jul 31 2005
%F A108891 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 A108891 Table begins
%e A108891   n\k  1    2    3    4    5    6
%e A108891   -------------------------------
%e A108891   1 |  2
%e A108891   2 |  2    4
%e A108891   3 |  6    8    8
%e A108891   4 | 22   28   24   16
%e A108891   5 | 90  112   96   64   32
%e A108891   6 |394  484  416  288  160   64
%e A108891 The paths DD, END, DEN, ENEN each have 2 returns (E=east, N=north, D=northeast); so T(2,2)=4.
%e A108891 From _Philippe Deléham_, Nov 02 2013: (Start)
%e A108891 Triangle (0, 1, 2, 1, 2, 1, 2, ...) DELTA (1, 0, 0, 0, ...) begins:
%e A108891   1;
%e A108891   0,   2;
%e A108891   0,   2,   4;
%e A108891   0,   6,   8,   8;
%e A108891   0,  22,  28,  24,  16;
%e A108891   0,  90, 112,  96,  64,  32;
%e A108891   0, 394, 484, 416, 288, 160,  64; (End)
%t A108891 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 A108891 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 A108891 nonn,tabl
%O A108891 1,1
%A A108891 _David Callan_, Jul 25 2005