A067304 - OEIS

%I #12 Jun 09 2019 21:15:23

%S 1,2,1,9,5,4,64,36,32,28,584,328,300,284,256,6144,3440,3184,3072,2960,

%T 2704,70576,39408,36704,35680,34896,33872,31168,859520,478912,447744,

%U 436928,429760,422592,411776

%N Generalized Catalan triangle A067298 with row reversion.

%C 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.

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

%F a(n, m) = A067298(n, n-m), n >= m >= 0, otherwise 0.

%F 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).

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

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

%K nonn,easy,tabl

%O 0,2

%A _Wolfdieter Lang_, Feb 05 2002