login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A125653 Triangle T, read by rows, where column k equals the eigensequence of matrix power T^k. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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: A131508 A140130 A220632 * A104445 A000189 A000190

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 21 18:11 EST 2014. Contains 252324 sequences.