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A255494
Triangle read by rows: coefficients of numerator of generating functions for powers of Pell numbers.
6
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
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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Mar 06 2015
STATUS
approved