A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

0, 0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 5, 9, 5, 0, 1, 7, 21, 27, 11, 0, 1, 9, 39, 85, 81, 21, 0, 1, 11, 63, 203, 341, 243, 43, 0, 1, 13, 93, 405, 1031, 1365, 729, 85, 0, 1, 15, 129, 715, 2511, 5187, 5461, 2187, 171, 0, 1, 17, 171, 1157, 5261, 15309, 25999, 21845, 6561, 341, 0, 1

Offset: 0

Author

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

Keywords

• nonn
• tabl

Comments

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 nondiagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = nondiagonal 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 antidiagonals.

Examples

			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, ...
  ...
		

Programs

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())