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

0,5

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

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.

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)

Example

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;

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 *)

Prog

(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]];
(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

Crossrefs

Cf. A000129, A079291, A094706, A110272.

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

Sequence in context: A139382 A157180 A179086 * A146956 A152613 A157153

Adjacent sequences: A255491 A255492 A255493 * A255495 A255496 A255497

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 06 2015

Status

approved