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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.) 3
2, 2, 4, 6, 8, 8, 22, 28, 24, 16, 90, 112, 96, 64, 32, 394, 484, 416, 288, 160, 64, 1806, 2200, 1896, 1344, 800, 384, 128, 8558, 10364, 8952, 6448, 4000, 2112, 896, 256, 41586, 50144, 43392, 31616, 20160, 11264, 5376, 2048, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Scott Balchin, Ethan MacBrough, and Kyle Ormsby, The combinatorics of N_oo operads for C_qp^n and D_p^n, arXiv:2209.06992 [math.AT], 2022.
FORMULA
Column k is the k-fold convolution of column 1.
T(n, k) = A104219(n-1, k-1)*2^k. - Philippe Deléham, Jul 31 2005
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
EXAMPLE
Table begins
n\k 1 2 3 4 5 6
-------------------------------
1 | 2
2 | 2 4
3 | 6 8 8
4 | 22 28 24 16
5 | 90 112 96 64 32
6 |394 484 416 288 160 64
The paths DD, END, DEN, ENEN each have 2 returns (E=east, N=north, D=northeast); so T(2,2)=4.
From Philippe Deléham, Nov 02 2013: (Start)
Triangle (0, 1, 2, 1, 2, 1, 2, ...) DELTA (1, 0, 0, 0, ...) begins:
1;
0, 2;
0, 2, 4;
0, 6, 8, 8;
0, 22, 28, 24, 16;
0, 90, 112, 96, 64, 32;
0, 394, 484, 416, 288, 160, 64; (End)
MATHEMATICA
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 *)
CROSSREFS
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).
Sequence in context: A194693 A308841 A176528 * A147570 A049625 A202102
KEYWORD
nonn,tabl
AUTHOR
David Callan, Jul 25 2005
STATUS
approved

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Last modified July 14 05:06 EDT 2024. Contains 374291 sequences. (Running on oeis4.)