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A067304 Generalized Catalan triangle A067298 with row reversion.

%I A067304 #21 Apr 21 2025 02:40:14
%S A067304 1,2,1,9,5,4,64,36,32,28,584,328,300,284,256,6144,3440,3184,3072,2960,
%T A067304 2704,70576,39408,36704,35680,34896,33872,31168,859520,478912,447744,
%U A067304 436928,429760,422592,411776,380608,10909440,6068480,5687872,5563200,5487488,5421952,5346240,5221568,4840960
%N A067304 Generalized Catalan triangle A067298 with row reversion.
%C A067304 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.
%F A067304 T(n, m) = A067298(n, n-m), n >= m >= 0, otherwise 0.
%F A067304 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 A067304 Triangle begins:
%e A067304     1;
%e A067304     2,   1;
%e A067304     9,   5,   4;
%e A067304    64,  36,  32,  28;
%e A067304   584, 328, 300, 284, 256;
%e A067304   ...
%e A067304 n=3: T(3, 0) = 64 = 36+28 = 32+32.
%Y A067304 The columns give for m=0..4: A067297 (diagonals of A067298), A067305, A067306, A067307, A067308.
%Y A067304 Cf. A067302 (row sums), A067323 (corresponding triangle for ordinary Catalan numbers).
%Y A067304 Cf. A000108, A067298, A067329.
%K A067304 nonn,tabl
%O A067304 0,2
%A A067304 _Wolfdieter Lang_, Feb 05 2002
%E A067304 More terms from _Jinyuan Wang_, Apr 20 2025