A067304 Generalized Catalan triangle A067298 with row reversion.

1, 2, 1, 9, 5, 4, 64, 36, 32, 28, 584, 328, 300, 284, 256, 6144, 3440, 3184, 3072, 2960, 2704, 70576, 39408, 36704, 35680, 34896, 33872, 31168, 859520, 478912, 447744, 436928, 429760, 422592, 411776, 380608, 10909440, 6068480, 5687872, 5563200, 5487488, 5421952, 5346240, 5221568, 4840960

Offset

0,2

Comments

Identity for each row n >= 1: T(n, m) + T(n, n-m+1) = A067297(n+1) (convolution of generalized Catalan numbers) for every m = 1..floor((n+1)/2). E.g., T(2*k+1, k+1) = A067297(2*(k+1))/2.

Links

Table of n, a(n) for n=0..44.

Formula

T(n, m) = A067298(n, n-m), n >= m >= 0, otherwise 0.

G.f. for column m >= 1 (without leading zeros): (2^(2*ceiling(m/2))*p(m, y)*(y^3)/(1+y)^4, where y = y(x) = c(4*x), with c(x) = g.f. of A000108 (Catalan) and the row polynomials p(n, y) = Sum_{k=0..n} A067329(n, k)*y^k, n >= 1. For m = 0: ((y*(3+y))^2)/(1+y)^4 with y = y(x) = c(4*x) (see A067297).

Example

Triangle begins:
    1;
    2,   1;
    9,   5,   4;
   64,  36,  32,  28;
  584, 328, 300, 284, 256;
  ...
n=3: T(3, 0) = 64 = 36+28 = 32+32.

Crossrefs

The columns give for m=0..4: A067297 (diagonals of A067298), A067305, A067306, A067307, A067308.

Cf. A067302 (row sums), A067323 (corresponding triangle for ordinary Catalan numbers).

Cf. A000108, A067298, A067329.

Sequence in context: A177972 A011136 A361146 * A103876 A133174 A155545

Adjacent sequences: A067301 A067302 A067303 * A067305 A067306 A067307

Keyword

nonn,tabl

Author

Wolfdieter Lang, Feb 05 2002

Extensions

More terms from Jinyuan Wang, Apr 20 2025

Status

approved