%I #14 Jan 24 2020 03:27:35
%S 1,2,1,2,3,2,2,5,7,3,2,7,14,13,5,2,9,23,32,25,8,2,11,34,62,71,46,13,2,
%T 13,47,105,156,149,84,21,2,15,62,163,295,367,304,151,34,2,17,79,238,
%U 505,767,827,604,269,55,2,19,98,332,805,1435,1889,1798,1177,475
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209127; see the Formula section.
%C u(n,n) = A000045(n), Fibonacci numbers.
%C Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,...
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle given by (1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 21 2012
%F u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F v(n,x) = x*u(n-1,x) + x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 21 2012: (Start)
%F As DELTA-triangle with 0 <= k <= n:
%F G.f.: (1-y*x+x^2-y^2*x^2)/(1-x-y*x-y^2*x^2).
%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e First five rows:
%e 1;
%e 2, 1;
%e 2, 3, 2;
%e 2, 5, 7, 3;
%e 2, 7, 14, 13, 5;
%e First three polynomials u(n,x):
%e 1
%e 2 + x
%e 2 + 3x + 2x^2
%e From _Philippe Deléham_, Mar 21 2012: (Start)
%e (1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 2, 1, 0;
%e 2, 3, 2, 0;
%e 2, 5, 7, 3, 0;
%e 2, 7, 14, 13, 5, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A209126 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A209127 *)
%Y Cf. A209127, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 05 2012