0,9

Consider a 3 X 3 matrix M =

[n, 1, 1]

[1, n, 1]

[1, 1, n].

The n-th row of the array contains the values of the non diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non diagonal entry + (n-1)^k.)

Table:

n: row sequence G.f. cross references.

0: (2^n-(-1)^n)/3 1/((1+1x)(1-2x)) A001045 (Jacobsthal sequence)

1: (3^n-0^n)/3 1/(1-3x)) A000244

2: (4^n-1^n)/3 1/((1-1x)(1-4x)) A002450

3: (5^n-2^n)/3 1/((1-2x)(1-5x)) A016127

4: (6^n-3^n)/3 1/((1-3x)(1-6x)) A016137

5: (7^n-4^n)/3 1/((1-4x)(1-7x)) A016150

6: (8^n-5^n)/3 1/((1-5x)(1-8x)) A016162

7: (9^n-6^n)/3 1/((1-6x)(1-9x)) A016172

8: (10^n-7^n)/3 1/((1-7x)(1-10x)) A016181

9: (11^n-8^n)/3 1/((1-8x)(1-11x)) A016187

10:(12^n-9^n)/3 1/((1-9x)(1-12x)) A016191

If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+2.

Columns follow polynomials with certain binomial coefficients:

column: polynomial

0; 0

1: 1

2: 2*n + 1

3: 3*n^2+ 3*n + 3

4: 4*n^3+ 6*n^2+ 12*n + 5

5: 5*n^4+10*n^3+ 30*n^2+ 25*n + 11

6: 6*n^5+15*n^4+ 60*n^3+ 75*n^2+ 66*n + 21

7: 7*n^6+21*n^5+105*n^4+ 175*n^3+ 231*n^2+ 147*n + 43

8: 8*n^7+28*n^6+168*n^5+ 350*n^4+ 616*n^3+ 588*n^2+344*n+ 85

etc.

Coefficients are generated by the array T(n,k)=(2^(n-k-1)-(-1)^(n-k-1))/3*(binomial(k+(n-k-1),n-k-1)) read by anti-diagonals.

Array begins:

0,1,1,3,5,11,...

0,1,3,9,27,81,...

0,1,5,21,85,341,...

0,1,7,39,203,1031,...

0,1,9,63,405,2511,...

...

(PARI) MM(n, N)=local(M); M=matrix(n, n); for(i=1, n, for(j=1, n, if(i==j, M[i, j]=N, M[i, j]=1))); M for(k=0, 10, for(i=0, 10, print1((MM(3, k)^i)[1, 2], ", ")); print())

Adjacent sequences: A102749 A102750 A102751 this_sequence A102753 A102754 A102755

Sequence in context: A021307 A129533 A060523 this_sequence A104548 A085707 A010607

nonn,tabl

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 09 2005