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Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the (n-1)-th diagonal with the k-th row of this triangle.
1

%I #7 Jun 13 2017 22:11:32

%S 1,1,1,1,2,1,1,3,4,1,1,4,9,9,1,1,5,16,30,24,1,1,6,25,70,115,77,1,1,7,

%T 36,135,344,510,295,1,1,8,49,231,805,1908,2602,1329,1,1,9,64,364,1616,

%U 5325,11904,15133,6934,1,1,10,81,540,2919,12381,39001,83028,99367,41351,1

%N Triangle, read by rows, such that T(n,k) equals the k-th term of the convolution of the (n-1)-th diagonal with the k-th row of this triangle.

%C The rows of this triangle are the reverse of the rows of triangle A091351, in which the k-th column lists the row sums of the k-th matrix power of A091351. Row sums form A091352 and equal the secondary diagonal.

%F T(n, k) = Sum_{i=0..k} T(k, i)*T(n-i-1, k-i) for 0<k<n, else T(0, n)=T(n, n)=1.

%e T(7,3) = T(3,0)*T(6,3) + T(3,1)*T(5,2) + T(3,2)*T(4,1) + T(3,3)*T(3,0)

%e = 1*70 + 3*16 + 4*4 + 1*1 = 135.

%e Rows begin:

%e [1],

%e [1,1],

%e [1,2,1],

%e [1,3,4,1],

%e [1,4,9,9,1],

%e [1,5,16,30,24,1],

%e [1,6,25,70,115,77,1],

%e [1,7,36,135,344,510,295,1],

%e [1,8,49,231,805,1908,2602,1329,1],

%e [1,9,64,364,1616,5325,11904,15133,6934,1],...

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

%Y Cf. A091351, A091352.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Sep 07 2004