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Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
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%I #8 Feb 19 2014 04:35:27

%S 1,0,2,0,2,3,0,2,7,4,0,2,11,16,5,0,2,15,36,30,6,0,2,19,64,91,50,7,0,2,

%T 23,100,204,196,77,8,0,2,27,144,385,540,378,112,9,0,2,31,196,650,1210,

%U 1254,672,156,10,0,2,35,256,1015,2366,3289,2640,1122,210,11

%N Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%C Row sums are A001519(n+1) = A122367(n).

%C Diagonal sums are A052969(n).

%F G.f.: (1-x)/(1-x-2*x*y+x^2*y^2).

%F Sum_{k=0..n} T(n,k)*2^k = 4^n = A000302(n).

%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.

%e Triangle begins:

%e 1;

%e 0, 2;

%e 0, 2, 3;

%e 0, 2, 7, 4;

%e 0, 2, 11, 16, 5;

%e 0, 2, 15, 36, 30, 6;

%e 0, 2, 19, 64, 91, 50, 7;

%e 0, 2, 23, 100, 204, 196, 77, 8;

%e 0, 2, 27, 144, 385, 540, 278, 112, 9;

%e 0, 2, 31, 196, 650, 1210, 1254, 672, 156, 10;

%e 0, 2, 35, 256, 1015, 2366, 3289, 2640, 1122, 210, 11;

%e ...

%Y Cf. A016742, A004767, A005581, A005583

%K nonn,tabl

%O 0,3

%A _Philippe Deléham_, Feb 18 2014