A067305 -id:A067305 - cp's OEIS frontend

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

Wolfdieter Lang, Feb 05 2002

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.

			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.

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.

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).

More terms from Jinyuan Wang, Apr 20 2025

cp's OEIS Frontend

A067304 Generalized Catalan triangle A067298 with row reversion.

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