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

Offset

0,2

Comments

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

The first column sequences (diagonals of A067298) are: A067297(n), A067305-8 for m=0..4.

Links

Table of n, a(n) for n=0..34.

Formula

a(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

{1}; {2,1}; {9,5,4}; {64,36,32,28}; ...; n=3: 64 = 36+28 = 32+32.

Crossrefs

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

Sequence in context: A177972 A011136 A361146 * A103876 A133174 A155545

Adjacent sequences: A067301 A067302 A067303 * A067305 A067306 A067307

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Feb 05 2002

Status

approved