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

0,5

Links

Table of n, a(n) for n=0..54.

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

Example

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

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A359759 A137431 A131222 * A114151 A243098 A360177

Adjacent sequences: A228331 A228332 A228333 * A228335 A228336 A228337

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