|
%I #9 Feb 09 2014 22:43:09
%S 1,1,2,4,5,9,28,32,36,64,256,284,300,328,584,2704,2960,3072,3184,3440,
%T 6144,31168,33872,34896,35680,36704,39408,70576,380608,411776,422592,
%U 429760,436928,447744,478912
%N Generalized Catalan triangle, based on C(2,2; n) := A064340(n).
%C For corresponding Catalan triangle with C(1,1; n) := A000108(n) see A028364.
%C Identity for each row n>=1: a(n,m)+a(n,n-(m+1))= a(n,n) = A067297(n) for m=0..floor((n-1)/2.). E.g., a(2k+1,k)= A067297(2*k+1)/2.
%C The columns (without leading zeros) give for m=0..3: A064340, A067299, 3*A067300, 8*A067301. The main diagonal gives A067297. The row sums give A067302.
%F a(n, m)= sum(C(2, 2; j)C(2, 2; n-j), j=0..m) if n>=m>=0 else 0.
%F 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(C(2, 2; n)*x^n, n=0..m) and c2(m, x) := sum(A067297(n)*x^n, n=0..m) for m=0, 1, 2, ...
%e {1}; {1,2}; {4,5,9}; {28,32,36,64}; ...
%K nonn,easy,tabl
%O 0,3
%A _Wolfdieter Lang_, Feb 05 2002
|
