A255494 Triangle read by rows: coefficients of numerator of generating functions for powers of Pell numbers.

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 38, 130, 38, 1, 1, 105, 1106, 1106, 105, 1, 1, 280, 8575, 26544, 8575, 280, 1, 1, 729, 62475, 567203, 567203, 62475, 729, 1, 1, 1866, 435576, 11179686, 32897774, 11179686, 435576, 1866, 1, 1, 4717, 2939208, 207768576, 1736613466, 1736613466, 207768576, 2939208, 4717, 1

Offset: 0

Author

N. J. A. Sloane, Mar 06 2015

Keywords

• nonn
• tabl

Comments

Note that Table 8 by Falcon should be labeled with the powers n (not r) and that the labels are off by 1. - R. J. Mathar, Jun 14 2015

Examples

			Triangle begins:
  1;
  1,    1; # see A079291
  1,    4,      1; # see A110272
  1,   13,     13,        1;
  1,   38,    130,       38,        1;
  1,  105,   1106,     1106,      105,        1;
  1,  280,   8575,    26544,     8575,      280,      1;
  1,  729,  62475,   567203,   567203,    62475,    729,    1;
  1, 1866, 435576, 11179686, 32897774, 11179686, 435576, 1866, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.

Crossrefs

Cf. A000129, A079291, A094706, A110272.

Diagonals: A094706, A255495, A255496, A255497, A255498.

Programs

Magma

P:= func< n | Round(((1 + Sqrt(2))^n - (1 - Sqrt(2))^n)/(2*Sqrt(2))) >;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return P(n-k+1)*T(n-1,k-1) + P(k+1)*T(n-1,k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]];

Mathematica

T[n_, k_]:= T[n,k]= If[k==0 || k==n, 1, Fibonacci[n-k+1, 2]*T[n-1, k-1] + Fibonacci[k+1, 2]*T[n-1, k]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 19 2021 *)

Sage

@CachedFunction
def P(n): return lucas_number1(n, 2, -1)
def T(n,k): return 1 if (k==0 or k==n) else P(n-k+1)*T(n-1, k-1) + P(k+1)*T(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 19 2021

Formula

From G. C. Greubel, Sep 19 2021: (Start)

T(n, k) = P(n-k+1)*T(n-1, k-1) + P(k+1)*T(n-1, k), where T(n, 0) = T(n, n) = 1 and P(n) = A000129(n).

T(n, k) = T(n, n-k).

T(n, 1) = A094706(n).

T(n, 2) = A255495(n-2).

T(n, 3) = A255496(n-3).

T(n, 4) = A255497(n-4).

T(n, 5) = A255498(n-5). (End)