0,6

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

...

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,...

...

(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()) p(n, k)=((n+4)^k-(n-1)^k)/5 for(k=0, 10, for(i=0, 10, print1(p(k, i), ", ")); print()) for(k=0, 10, for(i=0, 10, print1(polcoeff(x/((1-(k-1)*x)*(1-(k+4)*x)), i), ", ")); print())

Cf. A015521 (for n=0), A000351 (for n=1), A003464 (for n=2), A016130 (for n=3), A016140 (for n=4), A016153 (for n=5), A016164 (for n=6), A016174 (for n=7), A016184 (for n=8), A015441 (for n=-1), A091005 (for n=-2).

Sequence in context: A034261 A046778 A119925 this_sequence A129684 A105147 A111924

Adjacent sequences: A102762 A102763 A102764 this_sequence A102766 A102767 A102768

nonn,tabl

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 10 2005