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%I #29 Mar 09 2017 14:52:46
%S 1,1,1,1,1,2,1,1,3,3,1,1,4,5,5,1,1,5,7,10,8,1,1,6,9,16,18,13,1,1,7,11,
%T 23,31,33,21,1,1,8,13,31,47,62,59,34,1,1,9,15,40,66,101,119,105,55,1,
%U 1,10,17,50,88,151,205,227,185,89,1,1,11,19,61,113,213,321,414
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section.
%C Coefficient of x^(n-1): A000045(n) (Fibonacci numbers).
%C n-th row sum: 2^(n-1).
%C Mirror image of triangle in A053538. - _Philippe Deléham_, Mar 05 2012
%C Subtriangle of the triangle T(n,k) given by (1, 0, -1, 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 12 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 T(n,k) = A208747(n,k)/2^k. - _Philippe Deléham_, Mar 05 2012
%F From _Philippe Deléham_, Mar 12 2012: (Start)
%F As DELTA-triangle T(n,k) with 0<=k<=n:
%F G.f.: (1-y*x+y*x^2-y^2*x^2)/(1-x-y*x+t*x^2-y^2*x^2).
%F T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)
%F O.g.f.: 1/(1 - z - x*z(1 - z + x*z)) = 1 + (1 + x)*z + (1 + x + 2*x^2)*z^2 + (1 + x + 3*x + 3*x^2)*z^3 + .... - _Peter Bala_, Dec 31 2015
%F u(n,x) = Sum_{j=1..floor((n+1)/2)} (-1)^(j-1)*binomial(n-j,j-1)*(x*(1-x))^(j-1)* (1+x)^(n+1-2*j) for n>=1. - _Werner Schulte_, Mar 07 2017
%F T(n,k) = Sum_{j=0..floor((k-1)/2)} binomial(k-1-j,j)*binomial(n-k+j,j) for k,n>0 and k<=n (conjectured). - _Werner Schulte_, Mar 07 2017
%e First five rows:
%e 1
%e 1, 1
%e 1, 1, 2
%e 1, 1, 3, 3
%e 1, 1, 4, 5, 5
%e First five polynomials u(n,x): 1, 1 + x, 1 + x + x^2, 1 + x + 3*x^2 + 3*x^3, 1 + x + 4*x^2 + 5*x^3 + 5*x^4.
%e (1, 0, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
%e 1
%e 1, 0
%e 1, 1, 0
%e 1, 1, 2, 0
%e 1, 1, 3, 3, 0
%e 1, 1, 4, 5, 5, 0
%e 1, 1, 5, 7, 10, 8, 0
%e 1, 1, 6, 9, 16, 18, 13, 0
%e 1, 1, 7, 11, 23, 31, 33, 21, 0
%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, A208510.
%K nonn,tabl,easy
%O 1,6
%A _Clark Kimberling_, Feb 25 2012
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