cp's OEIS Frontend

A067298 Generalized Catalan triangle, based on C(2,2; n) = A064340(n).

%I A067298 #16 Apr 21 2025 02:40:51
%S A067298 1,1,2,4,5,9,28,32,36,64,256,284,300,328,584,2704,2960,3072,3184,3440,
%T A067298 6144,31168,33872,34896,35680,36704,39408,70576,380608,411776,422592,
%U A067298 429760,436928,447744,478912,859520,4840960,5221568,5346240,5421952,5487488,5563200,5687872,6068480,10909440
%N A067298 Generalized Catalan triangle, based on C(2,2; n) = A064340(n).
%C A067298 For corresponding Catalan triangle with C(1,1; n) = A000108(n) see A028364.
%C A067298 Identity for each row n>=1: T(n, m) + T(n, n-(m+1)) = T(n, n) = A067297(n) for m=0..floor((n-1)/2). E.g., T(2*k+1, k) = A067297(2*k+1)/2.
%F A067298 T(n, m) = Sum_{i=0..m} C(2,2; i)*C(2,2; n-i) if n >= m >= 0 else 0.
%F A067298 G.f. for column m (without leading zeros): (c(m, x)*c(2,2; x)-c2(m-1, x))/x^m, with c(2,2; x) = (1-3*x*c(4*x))/(1-2*x*c(4*x))^2 (g.f. for C(2,2; n)), c(x) = g.f. for Catalan numbers A000108, c(m, x) = Sum_{n=0..m} C(2,2; n)*x^n and c2(m, x) = Sum_{n=0..m} A067297(n)*x^n for m=0, 1, 2, ...
%e A067298 Triangle begins:
%e A067298     1;
%e A067298     1,   2;
%e A067298     4,   5,   9;
%e A067298    28,  32,  36,  64;
%e A067298   256, 284, 300, 328, 584;
%e A067298   ...
%o A067298 (PARI) A064340(n) = if(n>1, sum(m=0, n-2, (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)/2^(m+1))*(4^(n-1))/(n-1), 1);
%o A067298 T(n, m) = sum(i=0, m, A064340(i)*A064340(n-i)); \\ _Jinyuan Wang_, Apr 20 2025
%Y A067298 The columns (without leading zeros) give for m=0..3: A064340, A067299, 3*A067300, 8*A067301.
%Y A067298 The main diagonal gives A067297. The row sums give A067302.
%Y A067298 Cf. A000108, A067304.
%K A067298 nonn,tabl
%O A067298 0,3
%A A067298 _Wolfdieter Lang_, Feb 05 2002
%E A067298 More terms from _Jinyuan Wang_, Apr 20 2025