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%I #20 Jan 26 2020 21:07:45
%S 1,0,2,0,1,3,0,1,2,5,0,1,2,5,8,0,1,2,6,10,13,0,1,2,7,13,20,21,0,1,2,8,
%T 16,29,38,34,0,1,2,9,19,39,60,71,55,0,1,2,10,22,50,86,122,130,89,0,1,
%U 2,11,25,62,116,187,241,235,144,0,1,2,12,28,75,150,267,392,468
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.
%C u(n,n) = A000045(n+1) (Fibonacci numbers).
%C n-th row sum: 2^(n-1)
%C As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 26 2012
%F u(n,x) = u(n-1,x) + x*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_, Feb 26 2012: (Start)
%F As triangle T(n,k) with 0 <= k <= n:
%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k > n or if k < 0.
%F G.f.: (1-(1-y)*x)/(1-(1+y)*x+y*(1-y)*x^2).
%F Sum_{k=0..n} T(n,k)*x^k = (-1)*A091003(n+1), A152166(n), A000007(n), A000079(n), A055099(n), A152224(n) for x = -2, -1, 0, 1, 2, 3 respectively.
%F Sum_{k=0..n} T(n,k)*x^(n-k) = A087205(n), A140165(n+1), A016116(n+1), A000045(n+2), A000079(n), A122367(n), A006012(n), A052961(n), A154626(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively. (End)
%F T(n,k) = A208748(n,k)/2^k. - _Philippe Deléham_, Mar 05 2012
%e First five rows:
%e 1;
%e 0, 2;
%e 0, 1, 3;
%e 0, 1, 2, 5;
%e 0, 1, 2, 5, 8;
%e First five polynomials v(n,x):
%e 1
%e 2x
%e x + 3x^2
%e x + 2x^2 + 5x^3
%e x + 2x^2 + 5x^3 + 8x^4.
%t u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t u[n_, x_] := u[n - 1, x] + x*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[%] (* A208342 *)
%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[%] (* A208343 *)
%Y Cf. A208343.
%Y Cf. A084938, A000045, A000079.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, Feb 25 2012
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