%I #6 Aug 18 2017 11:46:03

%S 1,1,1,1,4,1,1,9,9,1,1,16,34,16,1,1,25,90,90,25,1,1,36,195,328,195,36,

%T 1,1,49,371,931,931,371,49,1,1,64,644,2240,3334,2240,644,64,1,1,81,

%U 1044,4788,9846,9846

%N Generalized Pascal triangle.

%C Consider the 1-parameter family of triangles with g.f. (1-x(1+y))/(1-2x(1+y)+x^2(1+k*x+y^2)). A007318 corresponds to k=2. A056241 corresponds to k=1. A124216 corresponds to k=0. Row sums are A006012. Diagonal sums are A124217.

%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry4/barry142.html">On a Generalization of the Narayana Triangle</a>, J. Int. Seq. 14 (2011) # 11.4.5

%F G.f.: (1-x(1+y))/(1-2x(1+y)+x^2(1+y^2)); Number triangle T(n,k)=sum{j=0..n, C(n,j)C(j,2(j-k))2^(j-k)}.

%F Equals 2*A001263 - A007318; (i.e. twice the Narayana triangle minus Pascal's triangle). - _Gary W. Adamson_, Jun 14 2007

%e Triangle begins

%e 1,

%e 1, 1,

%e 1, 4, 1,

%e 1, 9, 9, 1,

%e 1, 16, 34, 16, 1,

%e 1, 25, 90, 90, 25, 1,

%e 1, 36, 195, 328, 195, 36, 1,

%e 1, 49, 371, 931, 931, 371, 49, 1

%Y Cf. A001263.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Oct 19 2006