%I A120730
%S A120730 1,0,1,0,1,1,0,0,2,1,0,0,2,3,1,0,0,0,5,4,1,0,0,0,5,9,5,1,0,0,0,0,14,14,
%T A120730 6,1,0,0,0,0,14,28,20,7,1,0,0,0,0,0,42,48,27,8,1,0,0,0,0,0,42,90,75,35,
%U A120730 9,1
%N A120730 Another version of Catalan triangle A009766.
%C A120730 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 1, 
               -1, 0, 0, 1, -1, 0, 0, ...] DELTA [1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 
               0, -1, 1, ...] where DELTA is the operator defined in A084938.
%C A120730 Aerated version gives A165408. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 22 2009]
%F A120730 Sum_{k, 0<=k<=n}x^k*T(n,n-k)= A001405(n), A126087(n), A128386(n), A121724(n), 
               A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x=1,
               2,3,4,5,6,7,8,9 respectively .
%F A120730 T(2*n,n)= A000108(n); A000108 : Catalan numbers.
%F A120730 Sum_{k, 0<=k<=n}T(n,k)^2 = A000108(n) and Sum_{n, n>=k}T(n,k) = A000108(k+1). 
               [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2008]
%F A120730 Sum_{k, 0<=k<=n}T(n,k)^3 = A003161(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 18 2008]
%F A120730 Sum_{k, 0<=k<=n}T(n,k)^4 = A129123(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 18 2008]
%F A120730 Sum{k=0..n, T(n,k)*x^k}= A000007(n), A001405(n), A151281(n), A151162(n), 
               A151254(n), A156195(n), A156361(n), A156362(n), A156566(n), A156577(n) 
               for x=0,1,2,3,4,5,6,7,8,9 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Feb 10 2009]
%e A120730 As a triangle, this begins:
%e A120730 1;
%e A120730 0, 1;
%e A120730 0, 1, 1;
%e A120730 0, 0, 2, 1;
%e A120730 0, 0, 2, 3, 1;
%e A120730 0, 0, 0, 5, 4, 1;
%e A120730 0, 0, 0, 5, 9, 5, 1;
%e A120730 0, 0, 0, 0, 14, 14, 6, 1;
%Y A120730 Sequence in context: A058626 A122856 A055791 this_sequence A122851 A064301 
               A060701
%Y A120730 Adjacent sequences: A120727 A120728 A120729 this_sequence A120731 A120732 
               A120733
%K A120730 nonn,tabl
%O A120730 0,9
%A A120730 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 17 2006, corrected Sep 
               15 2006
%E A120730 Corrected formula . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 
               16 2008