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A108891
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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.)
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3
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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
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OFFSET
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1,1
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LINKS
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FORMULA
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Column k is the k-fold convolution of column 1.
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
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EXAMPLE
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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.
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)
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MATHEMATICA
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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 *)
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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STATUS
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approved
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