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 A236918 #17 Jun 14 2022 06:29:27
%S A236918 1,1,1,1,2,3,1,3,7,8,1,4,12,22,24,1,5,18,43,73,75,1,6,25,72,156,246,
%T A236918 243,1,7,33,110,283,564,844,808,1,8,42,158,465,1092,2046,2936,2742,1,
%U A236918 9,52,217,714,1906,4178,7449,10334,9458,1,10,63,288,1043,3096,7670,15904,27231,36736,33062
%N A236918 Triangle read by rows: Catalan triangle of the k-Fibonacci sequence.
%C A236918 Reversal of the Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A000958. - _Philippe Deléham_, Feb 10 2014
%C A236918 Row sums are in A109262. - _Philippe Deléham_, Feb 10 2014
%H A236918 G. C. Greubel, <a href="/A236918/b236918.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A236918 Sergio Falcon, <a href="http://dx.doi.org/10.4134/CKMS.2013.28.4.827">Catalan transform of the K-Fibonacci sequence</a>, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.
%F A236918 T(n, k) = coefficient of [x^k]( p(n, x) ), where p(n, x) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n-j)*x^(n-1)*Fibonacci(j, 1/x) with p(0, x) = 1 and Fibonacci(n, x) are the Fibonacci polynomials. - _G. C. Greubel_, Jun 14 2022
%e A236918 Triangle begins:
%e A236918 1;
%e A236918 1, 1;
%e A236918 1, 2, 3;
%e A236918 1, 3, 7, 8;
%e A236918 1, 4, 12, 22, 24;
%e A236918 1, 5, 18, 43, 73, 75;
%e A236918 1, 6, 25, 72, 156, 246, 243;
%e A236918 1, 7, 33, 110, 283, 564, 844, 808;
%e A236918 1, 8, 42, 158, 465, 1092, 2046, 2936, 2742;
%e A236918 1, 9, 52, 217, 714, 1906, 4178, 7449, 10334, 9458;
%e A236918 1, 10, 63, 288, 1043, 3096, 7670, 15904, 27231, 36736, 33062;
%e A236918 ... - Extended by _Philippe Deléham_, Feb 10 2014
%t A236918 P[n_, x_]:= P[n,x]= If[n==0, 1, Sum[(j/(2*n-j))*Binomial[2*n-j, n-j]*Fibonacci[j, 1/x] *x^(n-1), {j,0,n}]];
%t A236918 T[n_, k_]:= Coefficient[P[n, x], x, k];
%t A236918 Table[T[n, k], {n,10}, {k, 0, n-1}]//Flatten (* _G. C. Greubel_, Jun 14 2022 *)
%o A236918 (SageMath)
%o A236918 def f(n,x): return sum( binomial(n-j-1, j)*x^(n-2*j-1) for j in (0..(n-1)//2) )
%o A236918 def p(n,x):
%o A236918 if (n==0): return 1
%o A236918 else: return sum( (j/(2*n-j))*binomial(2*n-j, n-j)*x^(n-1)*f(j, 1/x) for j in (0..n) )
%o A236918 def A236918(n,k): return ( p(n,x) ).series(x, n+1).list()[k]
%o A236918 flatten([[A236918(n,k) for k in (0..n-1)] for n in (1..12)]) # _G. C. Greubel_, Jun 14 2022
%Y A236918 Diagonals give A000958, A114495.
%Y A236918 Cf. A109262 (row sums).
%K A236918 nonn,tabl
%O A236918 1,5
%A A236918 _N. J. A. Sloane_, Feb 09 2014
%E A236918 More terms from _Philippe Deléham_, Feb 09 2014