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%I #10 Jun 13 2017 21:51:30
%S 1,1,1,1,2,1,1,4,3,1,1,9,9,4,1,1,24,30,16,5,1,1,77,115,70,25,6,1,1,
%T 295,510,344,135,36,7,1,1,1329,2602,1908,805,231,49,8,1,1,6934,15133,
%U 11904,5325,1616,364,64,9,1,1,41351,99367,83028,39001,12381,2919,540,81,10,1
%N Triangle T, read by rows, such that T(n,k) equals the (n-k)-th row sum of T^k, where T^k is the k-th power of T as a lower triangular matrix.
%C Since T(n,0)=1 for n>=0, then the k-th column of the lower triangular matrix T equals the leftmost column of T^(k+1) for k>=0.
%F T(n, k) = sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n>=k>0 with T(n, 0)=1 (n>=0).
%F Equals SHIFT_UP(A104445), or A104445(n+1, k) = T(n, k) for n>=k>=0, where triangular matrix X=A104445 satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.
%e T(7,3) = 344 = 1*1 + 9*3 + 9*9 + 4*30 + 1*115
%e = T(4,0)*T(2,2) +T(4,1)*T(3,2) +T(4,2)*T(4,2) +T(4,3)*T(5,2) +T(4,4)*T(6,2).
%e Rows begin:
%e {1},
%e {1,1},
%e {1,2,1},
%e {1,4,3,1},
%e {1,9,9,4,1},
%e {1,24,30,16,5,1},
%e {1,77,115,70,25,6,1},
%e {1,295,510,344,135,36,7,1},
%e {1,1329,2602,1908,805,231,49,8,1},
%e {1,6934,15133,11904,5325,1616,364,64,9,1},...
%o (PARI) T(n,k)=if(k>n || n<0 || k<0,0,if(k==0 || k==n,1, sum(j=0,n-k,T(n-k,j)*T(j+k-1,k-1)););)
%Y Cf. A091352, A091353, A091354, A104445.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Jan 02 2004
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