%I A026729
%S A026729 1,0,1,0,1,1,0,0,2,1,0,0,1,3,1,0,0,0,3,4,1,0,0,0,1,6,5,1,0,0,0,0,4,10,
%T A026729 6,1,0,0,0,0,1,10,15,7,1,0,0,0,0,0,5,20,21,8,1,0,0,0,0,0,1,15,35,28,9,
%U A026729 1,0,0,0,0,0,0,6,35,56,36,10,1,0,0,0,0,0,0,1,21,70,84,45,11,1,0,0,0,0
%N A026729 Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, 
               k >= 0, read by antidiagonals.
%C A026729 The signed triangular matrix T(n,k)*(-1)^(n-k) is the inverse matrix 
               of the triangular Catalan convolution matrix A106566(n,k), n=k>=0, 
               with A106566(n,k) = 0 if n<k . - Philippe DELEHAM Aug 01 2005
%C A026729 As a number triangle : unsigned version of A109466 . [From Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Oct 26 2008]
%C A026729 A063967*A130595 as infinite lower triangular matrices . [From Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Dec 11 2008]
%C A026729 Modulo 2, this sequence becomes A106344 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 18 2008]
%D A026729 L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, 
               Discrete Applied Math., 34 (1991), 229-239.
%F A026729 As a number triangle, this is defined by : T(n,0) = 0^n, T(0,k) = 0^k, 
               T(n,k) = T(n-1,k-1) + Sum_{j, j>=0} = (-1)^j*T(n-1,k+j)*A000108(j) 
               for n>0 and k>0 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 
               07 2005
%F A026729 As a triangle read by rows, it is [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA 
               [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined 
               in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 
               2006
%F A026729 As a number triangle, this is defined by T(n, k)=sum{i=0..n, (-1)^(n+i)C(n, 
               i)C(i+k, i-k)} and is the Riordan array ( 1, x/(1+x) ). The row sums 
               of this triangle are F(n+1). - Paul Barry (pbarry(AT)wit.ie), Jun 
               21 2004
%F A026729 Sum_{k, 0<=k<=n}x^k*T(n,k)= A000007(n), A000045(n+1), A002605(n), A030195(n+1), 
               A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), 
               A057093(n) for n=0,1,2,3,4,5,6,7,8,9,10 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 16 2006
%F A026729 T(n,k)= A109466(n,k)*(-1)^(n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 11 2008]
%e A026729 Array begins
%e A026729 1 0 0 0 0 0 ...
%e A026729 1 1 0 0 0 0 ...
%e A026729 1 2 1 0 0 0 ...
%e A026729 1 3 3 1 0 0 ...
%e A026729 1 4 6 4 1 0 ...
%e A026729 As a triangle, this begins
%e A026729 1
%e A026729 0 1
%e A026729 0 1 1
%e A026729 0 0 2 1
%e A026729 0 0 1 3 1
%e A026729 0 0 0 3 4 1
%e A026729 0 0 0 1 6 5 1
%e A026729 ...
%Y A026729 The official entry for Pascal's triangle is A007318. See also A052553.
%Y A026729 Cf. A030528 (subtriangle for 1<=k<=n).
%Y A026729 Sequence in context: A061670 A108063 A164846 this_sequence A109466 A076833 
               A071676
%Y A026729 Adjacent sequences: A026726 A026727 A026728 this_sequence A026730 A026731 
               A026732
%K A026729 nonn,tabl,easy
%O A026729 0,9
%A A026729 N. J. A. Sloane (njas(AT)research.att.com), Jan 19 2003
%E A026729 More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 19 2003