OFFSET
0,6
COMMENTS
Consider a 5x5 matrix M =
[n, 1, 1, 1, 1]
[1, n, 1, 1, 1]
[1, 1, n, 1, 1]
[1, 1, 1, n, 1]
[1, 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.)
For row r we have polynomial ((r+4)^n-(r-1)^n)/5. Corresponding g.f.s: x/((1-(r-1)x)(1-(r+4)x))
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+4.
Triangle T(n, k) = (4^(n-k-1)-(-1)^(n-k-1))/5*(binomial(k+(n-k-1),n-k-1)) gives coefficients for polynomials for the columns of the array. First four polynomial are:
1
3 + 2*k
13 + 9*k + 3*k^2
51 + 52*k + 18*k^2 + 4*k^3
...
EXAMPLE
Array begins:
0, 1, 3, 13, 51, 205, ...
0, 1, 5, 25, 125, 625, ...
0, 1, 7, 43, 259, 1555, ...
0, 1, 9, 67, 477, 3355, ...
0, 1, 11, 97, 803, 6505, ...
...
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(5, k)^i)[1, 2], ", ")); print())
(PARI) p(n, k)=((n+4)^k-(n-1)^k)/5
for(k=0, 10, for(i=0, 10, print1(p(k, i), ", ")); print())
(PARI) for(k=0, 10, for(i=0, 10, print1(polcoeff(x/((1-(k-1)*x)*(1-(k+4)*x)), i), ", ")); print())
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 10 2005
STATUS
approved