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Riordan array (1/(1-3x),x(1-x)/(1-3x)^2).
2

%I #6 Jan 18 2014 06:40:59

%S 1,3,1,9,8,1,27,45,13,1,81,216,106,18,1,243,945,690,192,23,1,729,3888,

%T 3915,1574,303,28,1,2187,15309,20223,10941,2993,439,33,1,6561,58320,

%U 97524,67788,24598,5072,600,38,1,19683,216513,446148,385560,177498

%N Riordan array (1/(1-3x),x(1-x)/(1-3x)^2).

%C Row sums are A082685. Diagonal sums are A114196.

%F T(n, k)=sum{j=0..n, C(n, j)C(j+k, 2k)2^(j-k)}; T(n, k)=sum{j=0..n-k, C(k, j)C(n+k-j, 2k)(-1)^j*3^(n-k-j)}.

%F T(n,k) = 6*T(n-1,k) + T(n-1,k-1) - 9*T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 3, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Jan 18 2014

%e Triangle begins

%e 1;

%e 3, 1;

%e 9, 8, 1;

%e 27, 45, 13, 1;

%e 81, 216, 106, 18, 1;

%e 243, 945, 690, 192, 23, 1;

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Nov 16 2005