OFFSET
0,9
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.
EXAMPLE
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, ...
...
PROG
(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())
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 09 2005
STATUS
approved