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 A113716 Triangle, read by rows, where row n forms a polynomial in y=3*k that generates diagonal n as k=0,1,2,... for n>=0; thus T(n,k) = Sum_{j=0..n-k} T(n-k,j)*(3*k)^j, with T(n,0)=T(n,n)=1. 4

%I #5 Jun 13 2017 23:36:18

%S 1,1,1,1,4,1,1,22,7,1,1,157,61,10,1,1,1372,601,118,13,1,1,14008,6595,

%T 1495,193,16,1,1,161995,79981,20206,3001,286,19,1,1,2079994,1065589,

%U 291394,48685,5281,397,22,1,1,29268778,15495415,4492621,825313,100456

%N Triangle, read by rows, where row n forms a polynomial in y=3*k that generates diagonal n as k=0,1,2,... for n>=0; thus T(n,k) = Sum_{j=0..n-k} T(n-k,j)*(3*k)^j, with T(n,0)=T(n,n)=1.

%e Triangle begins:

%e 1;

%e 1,1;

%e 1,4,1;

%e 1,22,7,1;

%e 1,157,61,10,1;

%e 1,1372,601,118,13,1;

%e 1,14008,6595,1495,193,16,1;

%e 1,161995,79981,20206,3001,286,19,1;

%e 1,2079994,1065589,291394,48685,5281,397,22,1;

%e 1,29268778,15495415,4492621,825313,100456,8497,526,25,1; ...

%e where diagonals are generated by row polynomials:

%e T(6,5) = (1) + (1)*(3*5) = 16.

%e T(6,4) = (1) + (4)*(3*4) + (1)*(3*4)^2 = 193.

%e T(6,3) = (1) + (22)*(3*3) + (7)*(3*3)^2 + (1)*(3*3)^3 = 1495.

%e T(6,2) = (1) + (157)*(3*2) + (61)*(3*2)^2 + (10)*(3*2)^3 + (1)*(3*2)^4 = 6595.

%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k || k==0,1,sum(j=0,n-k,T(n-k,j)*(3*k)^j)))

%Y Cf. A091150, A113711, A113717 (column 1), A113718 (column 2), A113719 (row sums).

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 08 2005

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