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A096815 Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the two prior rows indexed by (n-k) and k. 5

%I

%S 1,1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,4,4,4,1,1,1,5,6,7,5,1,1,1,6,9,11,9,

%T 6,1,1,1,7,11,17,17,13,7,1,1,1,8,14,24,30,26,16,8,1,1,1,9,18,32,42,50,

%U 36,21,9,1,1,1,10,21,42,61,79,76,51,25,10,1,1,1,11,25,53,85,118,129,115,67

%N Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the two prior rows indexed by (n-k) and k.

%C Row sums are A096816.

%F T(n, k) = Sum_{j=0..min(n-k, k)} T(n-k, j)*T(k, k-j), for n>=k>=1, with T(n, 0)=T(n+1, 1)=T(n, n)=1 for n>=0.

%e T(7,4) = 11 = 4th term of the convolution of row (7-4) and row 4:

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

%e Rows begin:

%e [1],

%e [1,1],

%e [1,1,1],

%e [1,1,2,1],

%e [1,1,3,3,1],

%e [1,1,4,4,4,1],

%e [1,1,5,6,7,5,1],

%e [1,1,6,9,11,9,6,1],

%e [1,1,7,11,17,17,13,7,1],

%e [1,1,8,14,24,30,26,16,8,1],

%e [1,1,9,18,32,42,50,36,21,9,1],

%e [1,1,10,21,42,61,79,76,51,25,10,1],

%e [1,1,11,25,53,85,118,129,115,67,31,11,1],...

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

%o (Maxima) T(n,k):= if ( n<k or k<0 ) then 0 else

%o if ( k<=1 or k=n ) then 1 else sum(T(n-k,j)*T(k,k-j),j,0, k);

%o create_list(T(n,k),n,0,12,k,0,n); /* _Emanuele Munarini_, May 12 2012 */

%Y Cf. A096816, A096811.

%K nonn,tabl

%O 0,9

%A _Paul D. Hanna_, Jul 20 2004

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