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A096589 Symmetric square array T(n,k)=T(k,n), read by antidiagonals, such that T(n,k) equals the dot product of the k-th antidiagonal with the initial terms of the (n-k)-th row when n>=k, with T(n,0)=1. 1

%I

%S 1,1,1,1,2,1,1,3,3,1,1,4,4,4,1,1,5,8,8,5,1,1,6,11,8,11,6,1,1,7,17,20,

%T 20,17,7,1,1,8,22,30,14,30,22,8,1,1,9,30,45,42,42,45,30,9,1,1,10,37,

%U 69,72,28,72,69,37,10,1,1,11,47,100,101,98,98,101,100,47,11,1,1,12,56,133,159

%N Symmetric square array T(n,k)=T(k,n), read by antidiagonals, such that T(n,k) equals the dot product of the k-th antidiagonal with the initial terms of the (n-k)-th row when n>=k, with T(n,0)=1.

%C Main diagonal equals the antidiagonal sums (A096590).

%F T(n, k) = Sum_{j=0, k} T(k-j, j)*T(n-k, j) when n>=k, else T(n, k)=T(k, n).

%e T(5,2) = 17 = 1*1+2*4+1*8 = T(2,0)*T(3,0) + T(1,1)*T(3,1) +

%e T(0,2)*T(3,2).

%e T(7,3) = 69 = 1*1+3*5+3*11+1*20 = T(3,0)*T(4,0) + T(2,1)*T(4,1) +

%e T(1,2)*T(4,2) + T(0,3)*T(4,3).

%e Rows begin:

%e [1,1,1,1,1,1,1,1,1,1,...],

%e [1,2,3,4,5,6,7,8,9,10,...],

%e [1,3,4,8,11,17,22,30,37,47,...],

%e [1,4,8,8,20,30,45,69,100,133,...],

%e [1,5,11,20,14,42,72,101,159,255,...],

%e [1,6,17,30,42,28,98,184,279,386,...],

%e [1,7,22,45,72,98,44,176,372,622,...],

%e [1,8,30,69,101,184,176,90,405,943,...],

%e [1,9,37,100,159,279,372,405,136,680,...],

%e [1,10,47,133,255,386,622,943,680,254,...],...

%o (PARI) T(n,k)=if(n<0 || k<0,0,if(n==0 || k==0,1, if(n>k, sum(j=0,k,T(n-k,j)*T(k-j,j)), sum(j=0,n,T(k-n,j)*T(n-j,j)););))

%Y Cf. A096590.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Jun 28 2004

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