A125653 - OEIS

%I #6 Jun 14 2017 00:14:54

%S 1,1,1,1,1,1,1,2,1,1,1,4,3,1,1,1,9,8,4,1,1,1,24,24,13,5,1,1,1,75,84,

%T 47,19,6,1,1,1,269,335,195,79,26,7,1,1,1,1091,1495,908,372,121,34,8,1,

%U 1,1,4940,7381,4674,1947,631,174,43,9,1,1,1,24699,39912,26327,11177,3632

%N Triangle T, read by rows, where column k equals the eigensequence of matrix power T^k.

%F T(n,k) = Sum_{j=0..n-k-1} [T^k](n-k-1,j)*T(j+k,k) for n>k>=0 with T(n,n)=1 for n>=0, where [T^k] denotes the matrix k power of T.

%e Triangle T begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 1, 1;

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

%e 1, 9, 8, 4, 1, 1;

%e 1, 24, 24, 13, 5, 1, 1;

%e 1, 75, 84, 47, 19, 6, 1, 1;

%e 1, 269, 335, 195, 79, 26, 7, 1, 1;

%e 1, 1091, 1495, 908, 372, 121, 34, 8, 1, 1;

%e 1, 4940, 7381, 4674, 1947, 631, 174, 43, 9, 1, 1;

%e 1, 24699, 39912, 26327, 11177, 3632, 989, 239, 53, 10, 1, 1; ...

%e Column 1 is the eigensequence of this triangle T:

%e T(5,1) = 1*(1) + 2*(1) + 1*(2) + 1*(4) = 9;

%e T(6,1) = 1*(1) + 4*(1) + 3*(2) + 1*(4) + 1*(9) = 24;

%e T(7,1) = 1*(1) + 9*(1) + 8*(2) + 4*(4) + 1*(9) + 1*(24) = 75.

%e Matrix square T^2 begins:

%e 1;

%e 2, 1;

%e 3, 2, 1;

%e 5, 5, 2, 1;

%e 10, 13, 7, 2, 1;

%e 24, 38, 23, 9, 2, 1;

%e 69, 127, 84, 35, 11, 2, 1 ...

%e Column 2 of T is the eigensequence of matrix square T^2:

%e T(5,2) = 3*(1) + 2*(1) + 1*(3) = 8;

%e T(6,2) = 5*(1) + 5*(1) + 2*(3) + 1*(8) = 24;

%e T(7,2) = 10*(1) + 13*(1) + 7*(3) + 2*(8) + 1*(24) = 84.

%e Matrix cube T^3 begins:

%e 1;

%e 3, 1;

%e 6, 3, 1;

%e 13, 9, 3, 1;

%e 33, 28, 12, 3, 1;

%e 97, 96, 46, 15, 3, 1;

%e 329, 367, 192, 67, 18, 3, 1 ...

%e Column 3 of T is the eigensequence of matrix cube T^3:

%e T(6,3) = 6*(1) + 3*(1) + 1*(4) = 13;

%e T(7,3) = 13*(1) + 9*(1) + 3*(4) + 1*(13) = 47;

%e T(8,3) = 33*(1) + 28*(1) + 12*(4) + 3*(13) + 1*(47) = 195.

%o (PARI) T(n,k)=local(M);M=matrix(n,n,r,c,if(r>=c,T(r-1,c-1))); if(n<k || k<0,0,if(n==k,1,sum(j=k,n-1,(M^k)[n-k,j-k+1]*T(j,k))))

%Y Rows: A125654, A125655, A125656, A125657; A125658 (row sums); A125659 (central terms).

%K nonn,tabl

%O 0,8

%A _Paul D. Hanna_, Nov 30 2006

%E Example corrected by _Paul D. Hanna_, Oct 29 2010