A255494 - OEIS

%I #34 Nov 17 2022 09:17:17

%S 1,1,1,1,4,1,1,13,13,1,1,38,130,38,1,1,105,1106,1106,105,1,1,280,8575,

%T 26544,8575,280,1,1,729,62475,567203,567203,62475,729,1,1,1866,435576,

%U 11179686,32897774,11179686,435576,1866,1,1,4717,2939208,207768576,1736613466,1736613466,207768576,2939208,4717,1

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

%C 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

%H G. C. Greubel, <a href="/A255494/b255494.txt">Rows n = 0..50 of the triangle, flattened</a>

%H S. Falcon, <a href="http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.

%F From _G. C. Greubel_, Sep 19 2021: (Start)

%F 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).

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

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

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

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

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

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

%e Triangle begins:

%e   1;

%e   1,    1; # see A079291

%e   1,    4,      1; # see A110272

%e   1,   13,     13,        1;

%e   1,   38,    130,       38,        1;

%e   1,  105,   1106,     1106,      105,        1;

%e   1,  280,   8575,    26544,     8575,      280,      1;

%e   1,  729,  62475,   567203,   567203,    62475,    729,    1;

%e   1, 1866, 435576, 11179686, 32897774, 11179686, 435576, 1866, 1;

%t 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 *)

%o (Magma)

%o P:= func< n | Round(((1 + Sqrt(2))^n - (1 - Sqrt(2))^n)/(2*Sqrt(2))) >;

%o function T(n,k)

%o   if k eq 0 or k eq n then return 1;

%o   else return P(n-k+1)*T(n-1,k-1) + P(k+1)*T(n-1,k);

%o   end if; return T;

%o end function;

%o [T(n,k): k in [0..n], n in [0..12]];

%o (Sage)

%o @CachedFunction

%o def P(n): return lucas_number1(n, 2, -1)

%o 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)

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 19 2021

%Y Cf. A000129, A079291, A094706, A110272.

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

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_, Mar 06 2015