A104445 - OEIS

A104445

Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT_UP(T) = T^2 - T + I, or, equivalently: T(n+1,k+1) = [T^2](n,k) - T(n,k) + [T^0](n,k) for n>=k>=0, with T(0,0)=1.

%I #6 Jun 13 2017 01:29:49

%S 1,1,1,1,1,1,1,2,1,1,1,4,3,1,1,1,9,9,4,1,1,1,24,30,16,5,1,1,1,77,115,

%T 70,25,6,1,1,1,295,510,344,135,36,7,1,1,1,1329,2602,1908,805,231,49,8,

%U 1,1,1,6934,15133,11904,5325,1616,364,64,9,1,1,1,41351,99367,83028,39001

%N Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT_UP(T) = T^2 - T + I, or, equivalently: T(n+1,k+1) = [T^2](n,k) - T(n,k) + [T^0](n,k) for n>=k>=0, with T(0,0)=1.

%C Surprisingly, SHIFT_UP(T) = A091351, or T(n+1,k) = A091351(n,k) for n>=k>=0, where column k of A091351 equals column 0 of A091351^(k+1) for k>=0.

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

%e Rows begin:

%e 1;

%e 1,1;

%e 1,1,1;

%e 1,2,1,1;

%e 1,4,3,1,1;

%e 1,9,9,4,1,1;

%e 1,24,30,16,5,1,1;

%e 1,77,115,70,25,6,1,1;

%e 1,295,510,344,135,36,7,1,1;

%e 1,1329,2602,1908,805,231,49,8,1,1;

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

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

%Y Cf. A091351, A104446 (matrix square); columns form: A091352, A091353, A091354.

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Mar 07 2005