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A228335
Triangle read by rows: the Y-transformation of the Catalan triangle A033184.
3
1, 1, 1, 3, 6, 1, 14, 40, 15, 1, 84, 300, 175, 28, 1, 594, 2475, 1925, 504, 45, 1, 4719, 22022, 21021, 7644, 1155, 66, 1, 40898, 208208, 231868, 107016, 23100, 2288, 91, 1, 379236, 2068560, 2598960, 1439424, 403920, 58344, 4095, 120, 1, 3711916, 21414900, 29651400, 18976896, 6523308, 1247103, 129675, 6800, 153, 1
OFFSET
0,4
COMMENTS
From Roger Ford, Feb 08 2020: (Start)
For 2n-step octant walks, the numbers for the x axis ending locations are equal to numbers in row n of the example triangle.
Example: For n = 2, all 4-step octant walks starting at (0,0) and ending on the x axis are as follows:
EWEW EEWW EEEW EWEE EEWE EEEE
ENSW EENS ENSE ENES
(0,0) (2,0) (4,0) - x axis ending location
3 6 1 - number of walks
These numbers match row 2 of the example triangle. (End)
LINKS
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013.
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.
EXAMPLE
Triangle begins:
1;
1, 1;
3, 6, 1;
14, 40, 15, 1;
84, 300, 175, 28, 1;
594, 2475, 1925, 504, 45, 1;
...
MATHEMATICA
nn = 9;
c[n_, k_] := Binomial[2n-k, n] (k+1)/(n+1);
a[0, 0] = 1;
a[n_, k_] := Table[c[n+k+i, 2k+j], {i, 2}, {j, 2}] // Det;
Table[a[n, k], {n, 0, nn}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)
PROG
(PARI) C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
T(n, k) = matdet(matrix(2, 2, i, j, C(n+k+i, 2*k+j))); \\ Michel Marcus, Feb 13 2014
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Aug 26 2013
EXTENSIONS
More terms from Michel Marcus, Feb 13 2014
A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014
STATUS
approved