A255494 - OEIS

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

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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; # 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)