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%I #62 Feb 11 2021 03:57:28
%S 1,1,1,1,2,2,1,3,5,3,1,4,9,10,5,1,5,14,22,20,8,1,6,20,40,51,38,13,1,7,
%T 27,65,105,111,71,21,1,8,35,98,190,256,233,130,34,1,9,44,140,315,511,
%U 594,474,235,55,1,10,54,192,490,924,1295,1324,942,420,89,1,11,65,255
%N Reflection of A037027: T(n,m) = U(n,n-m), m=0..n, where U is as in A037027.
%C Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (2,2). - _Joerg Arndt_, Jul 01 2011
%C The n-th diagonal D(n) = {T(n,0), T(n+1,1), ..., T(n+m,m), ...} of the triangle has generating function F(x) = 1/(1 - x - x^2)^(n+1) for n = 0,1,2,.... - _L. Edson Jeffery_, Mar 20 2011
%C Let p(n,x) denote the Fibonacci polynomial, defined by p(1,x) = 1, p(2,x) = x, and p(n,x) = x*p(n-1,x) + p(n-2,x) for n >= 3. Let q(n,x) be the numerator polynomial of the rational function p(n, 1 + 1/x). The coefficients of the polynomial q(n,x) are given by the (n-1)-th row of T(n,k). E.g., p(5,x) = 1 + 3*x^2 + x^4 gives q(5,x) = 1 + 4*x + 9*x^2 + 10*x^2 + 5*x^4. - _Clark Kimberling_, Nov 04 2013
%H Reinhard Zumkeller, <a href="/A038137/b038137.txt">Rows n = 0..150 of triangle, flattened</a>
%H Pieter Moree, <a href="https://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.
%H Pieter Moree, <a href="http://emis.impa.br/EMIS/journals/JIS/VOL7/Moree/moree12.html">Convoluted convolved Fibonacci numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciPolynomial.html">Fibonacci polynomial</a>.
%F From _Paul Barry_, Oct 24 2005: (Start)
%F G.f.: 1/(1 - x - x*y - x^2*y^2).
%F T(n,k) = Sum_{j=0..n} C((n+j)/2, j) * (1 + (-1)^(n+j)) * C(j, n-k)/2. (End)
%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(n,k) = 0 if n < 0 or if n < k, and T(0,0) = 1. - _Philippe Deléham_, Nov 30 2006
%F Sum_{k=0..n} (-1)^k*T(n,k) = A059841(n). - _Philippe Deléham_, Nov 30 2006
%F T(n,k) = A208336(n+1,k).- _Philippe Deléham_, Apr 05 2012
%e Triangle T(n,k) (with rows n >= 0 and columns 0 <= k <= n) begins
%e 1;
%e 1, 1;
%e 1, 2, 2;
%e 1, 3, 5, 3;
%e 1, 4, 9, 10, 5;
%e 1, 5, 14, 22, 20, 8;
%e 1, 6, 20, 40, 51, 38, 13;
%e 1, 7, 27, 65, 105, 111, 71, 21;
%e ...
%o (Haskell)
%o a038137 n k = a038137_tabl !! n !! k
%o a038137_row n = a038137_tabl !! n
%o a038137_tabl = map reverse a037027_tabl
%o -- _Reinhard Zumkeller_, Jul 08 2012
%Y Row sums are Pell numbers A000129.
%Y Diagonal sums are unsigned version of A077930.
%Y Cf. A037027, A059841, A208336.
%K easy,nonn,tabl
%O 0,5
%A _Floor van Lamoen_
%E Title corrected by _L. Edson Jeffery_, Apr 23 2011
%E Corrected by _Philippe Deléham_, Apr 05 2012
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