A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

0, 0, 1, 0, 1, 3, 0, 1, 5, 13, 0, 1, 7, 25, 51, 0, 1, 9, 43, 125, 205, 0, 1, 11, 67, 259, 625, 819, 0, 1, 13, 97, 477, 1555, 3125, 3277, 0, 1, 15, 133, 803, 3355, 9331, 15625, 13107, 0, 1, 17, 175, 1261, 6505, 23517, 55987, 78125, 52429, 0, 1, 19, 223, 1875, 11605

Offset: 0

Author

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

Keywords

• nonn
• tabl

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

...

Examples

			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, ...
  ...
		

Crossrefs

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).

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(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())