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%I #13 Aug 12 2015 04:03:53
%S 1,0,2,0,2,3,0,2,5,5,0,2,7,12,8,0,2,9,21,25,13,0,2,11,32,53,50,21,0,2,
%T 13,45,94,124,96,34,0,2,15,60,150,250,273,180,55,0,2,17,77,223,445,
%U 617,577,331,89,0,2,19,96,315,728,1212,1444,1181,600,144,0,2,21
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209126; see the Formula section.
%C u(n,n)=(1,2,3,5,8,13,21,...)=A000045(n+1), Fibonacci numbers.
%C Alternating row sums: (1,-2,1,-2,1,-2,1,-2,...
%C For a discussion and guide to related arrays, see A208510.
%C As triangle T(n,k) with 0<=k<=n, it is (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 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 T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 21 2012
%F G.f.: (-1-x*y+x)*x*y/(-1+x*y+x+x^2*y^2). - _R. J. Mathar_, Aug 12 2015
%e First five rows:
%e 1
%e 0...2
%e 0...2...3
%e 0...2...5...5
%e 0...2...7...12...8
%e First three polynomials v(n,x): 1, 2x, 2x + 3x^2.
%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. A209126, A208510.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Mar 05 2012
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