This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A255494 #34 Nov 17 2022 09:17:17
%S A255494 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 A255494 26544,8575,280,1,1,729,62475,567203,567203,62475,729,1,1,1866,435576,
%U A255494 11179686,32897774,11179686,435576,1866,1,1,4717,2939208,207768576,1736613466,1736613466,207768576,2939208,4717,1
%N A255494 Triangle read by rows: coefficients of numerator of generating functions for powers of Pell numbers.
%C A255494 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 A255494 G. C. Greubel, <a href="/A255494/b255494.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A255494 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 A255494 From _G. C. Greubel_, Sep 19 2021: (Start)
%F A255494 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 A255494 T(n, k) = T(n, n-k).
%F A255494 T(n, 1) = A094706(n).
%F A255494 T(n, 2) = A255495(n-2).
%F A255494 T(n, 3) = A255496(n-3).
%F A255494 T(n, 4) = A255497(n-4).
%F A255494 T(n, 5) = A255498(n-5). (End)
%e A255494 Triangle begins:
%e A255494 1;
%e A255494 1, 1; # see A079291
%e A255494 1, 4, 1; # see A110272
%e A255494 1, 13, 13, 1;
%e A255494 1, 38, 130, 38, 1;
%e A255494 1, 105, 1106, 1106, 105, 1;
%e A255494 1, 280, 8575, 26544, 8575, 280, 1;
%e A255494 1, 729, 62475, 567203, 567203, 62475, 729, 1;
%e A255494 1, 1866, 435576, 11179686, 32897774, 11179686, 435576, 1866, 1;
%t A255494 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 A255494 (Magma)
%o A255494 P:= func< n | Round(((1 + Sqrt(2))^n - (1 - Sqrt(2))^n)/(2*Sqrt(2))) >;
%o A255494 function T(n,k)
%o A255494 if k eq 0 or k eq n then return 1;
%o A255494 else return P(n-k+1)*T(n-1,k-1) + P(k+1)*T(n-1,k);
%o A255494 end if; return T;
%o A255494 end function;
%o A255494 [T(n,k): k in [0..n], n in [0..12]];
%o A255494 (Sage)
%o A255494 @CachedFunction
%o A255494 def P(n): return lucas_number1(n, 2, -1)
%o A255494 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 A255494 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 19 2021
%Y A255494 Cf. A000129, A079291, A094706, A110272.
%Y A255494 Diagonals: A094706, A255495, A255496, A255497, A255498.
%K A255494 nonn,tabl
%O A255494 0,5
%A A255494 _N. J. A. Sloane_, Mar 06 2015