A125653 Triangle T, read by rows, where column k equals the eigensequence of matrix power T^k.

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, 47, 19, 6, 1, 1, 1, 269, 335, 195, 79, 26, 7, 1, 1, 1, 1091, 1495, 908, 372, 121, 34, 8, 1, 1, 1, 4940, 7381, 4674, 1947, 631, 174, 43, 9, 1, 1, 1, 24699, 39912, 26327, 11177, 3632

Offset

0,8

Links

Table of n, a(n) for n=0..71.

Formula

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.

Example

Triangle T begins:
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, 47, 19, 6, 1, 1;
1, 269, 335, 195, 79, 26, 7, 1, 1;
1, 1091, 1495, 908, 372, 121, 34, 8, 1, 1;
1, 4940, 7381, 4674, 1947, 631, 174, 43, 9, 1, 1;
1, 24699, 39912, 26327, 11177, 3632, 989, 239, 53, 10, 1, 1; ...
Column 1 is the eigensequence of this triangle T:
T(5,1) = 1*(1) + 2*(1) + 1*(2) + 1*(4) = 9;
T(6,1) = 1*(1) + 4*(1) + 3*(2) + 1*(4) + 1*(9) = 24;
T(7,1) = 1*(1) + 9*(1) + 8*(2) + 4*(4) + 1*(9) + 1*(24) = 75.
Matrix square T^2 begins:
1;
2, 1;
3, 2, 1;
5, 5, 2, 1;
10, 13, 7, 2, 1;
24, 38, 23, 9, 2, 1;
69, 127, 84, 35, 11, 2, 1 ...
Column 2 of T is the eigensequence of matrix square T^2:
T(5,2) = 3*(1) + 2*(1) + 1*(3) = 8;
T(6,2) = 5*(1) + 5*(1) + 2*(3) + 1*(8) = 24;
T(7,2) = 10*(1) + 13*(1) + 7*(3) + 2*(8) + 1*(24) = 84.
Matrix cube T^3 begins:
1;
3, 1;
6, 3, 1;
13, 9, 3, 1;
33, 28, 12, 3, 1;
97, 96, 46, 15, 3, 1;
329, 367, 192, 67, 18, 3, 1 ...
Column 3 of T is the eigensequence of matrix cube T^3:
T(6,3) = 6*(1) + 3*(1) + 1*(4) = 13;
T(7,3) = 13*(1) + 9*(1) + 3*(4) + 1*(13) = 47;
T(8,3) = 33*(1) + 28*(1) + 12*(4) + 3*(13) + 1*(47) = 195.

Prog

(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))))

Crossrefs

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

Sequence in context: A354090 A362826 A220632 * A104445 A359762 A000189

Adjacent sequences: A125650 A125651 A125652 * A125654 A125655 A125656

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 30 2006

Extensions

Example corrected by Paul D. Hanna, Oct 29 2010

Status

approved