A228337 -id:A228337 - cp's OEIS frontend

A228334 Triangle read by rows: the X-transformation of the Catalan triangle A033184.

1, 0, 1, 0, 3, 1, 0, 14, 10, 1, 0, 84, 90, 21, 1, 0, 594, 825, 308, 36, 1, 0, 4719, 7865, 4004, 780, 55, 1, 0, 40898, 78078, 49686, 13650, 1650, 78, 1, 0, 379236, 804440, 606424, 214200, 37400, 3094, 105, 1, 0, 3711916, 8565960, 7379904, 3162816, 724812, 88179, 5320, 136, 1

Offset: 0

N. J. A. Sloane, Aug 26 2013

			Triangle begins:
  1;
  0,   1;
  0,   3,   1;
  0,  14,  10,   1;
  0,  84,  90,  21,   1;
  0, 594, 825, 308,  36,   1;
  ...

Fangfang Cai, Qing-Hu Hou, Yidong Sun, Arthur L.B. Yang, Combinatorial identities related to 2×2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018.
Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv preprint arXiv:1305.2015 [math.CO], 2013.
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

Cf. A033184, A179898, A228335, A228336, A228337.

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-1, 2k+j-1], {i, 1, 2}, {j, 1, 2}] // Det;
Table[a[n, k], {n, 0, nn}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
aX(nn) = {for (n = 0, nn, for (k = 0, n, print1(matdet(matrix(2, 2, i, j, C(n+k+i-1, 2*k+j-1))), ", ");); print(););} \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

A228335 Triangle read by rows: the Y-transformation of the Catalan triangle A033184.

Original entry on oeis.org

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

N. J. A. Sloane, Aug 26 2013

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)

			Triangle begins:
    1;
    1,    1;
    3,    6,    1;
   14,   40,   15,    1;
   84,  300,  175,   28,    1;
  594, 2475, 1925,  504,   45,    1;
  ...

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.

Cf. A033184, A228334, A228336, A228337.

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 *)

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

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 15, 12, 4, 1, 25, 45, 36, 20, 5, 1, 70, 126, 126, 70, 30, 6, 1, 196, 392, 392, 280, 120, 42, 7, 1, 588, 1176, 1344, 960, 540, 189, 56, 8, 1, 1764, 3780, 4320, 3600, 2025, 945, 280, 72, 9, 1, 5544, 11880, 14850, 12375, 8250, 3850, 1540, 396, 90, 10, 1

Offset: 0

N. J. A. Sloane, Aug 26 2013

			Triangle begins:
   1;
   1,   1;
   2,   2,   1;
   4,   6,   3,  1;
  10,  15,  12,  4,  1;
  25,  45,  36, 20,  5, 1;
  70, 126, 126, 70, 30, 6, 1;
  ...

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.

Cf. A033184, A228334, A228335, A228337.

c[n_, k_] := Boole[k <= n] Binomial[2n - k, n] (k + 1)/(n + 1);
T[n_, k_] := Module[{nn, kk}, If[OddQ[n], nn = (n + 1)/2, nn = n/2]; If[OddQ[k], kk = (k - 1)/2, kk = k/2]; If [OddQ[n], If[OddQ[k], c[nn + kk, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk, 2kk + 1]], If[OddQ[k], c[nn + kk + 1, 2kk + 1] c[nn + kk + 1, 2kk + 2], c[nn + kk, 2kk] c[nn + kk + 1, 2kk + 1]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 04 2018, from PARI *)

C(n, k) = (k<=n)*binomial(2*n-k, n)*(k+1)/(n+1);
T(n, k) = {my(nn, kk); if (n % 2, nn = (n+1)/2, nn = n/2); if (k % 2, kk = (k-1)/2, kk = k/2); if ((n % 2), if (k % 2, C(nn+kk, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk, 2*kk+1)), if (k % 2, C(nn+kk+1, 2*kk+1)*C(nn+kk+1, 2*kk+2), C(nn+kk, 2*kk)*C(nn+kk+1, 2*kk+1)));} \\ Michel Marcus, Feb 13 2014

More terms from Michel Marcus, Feb 13 2014

A-number for Catalan triangle changed by Michel Marcus, Feb 13 2014

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A228334 Triangle read by rows: the X-transformation of the Catalan triangle A033184.

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A228335 Triangle read by rows: the Y-transformation of the Catalan triangle A033184.

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A228336 Triangle read by rows: the Z-transformation of the Catalan triangle A033184.

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