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%I #12 Mar 30 2012 18:58:13
%S 1,1,1,1,3,2,1,6,7,3,1,10,16,14,5,1,15,30,40,28,8,1,21,50,90,93,53,13,
%T 1,28,77,175,238,203,99,21,1,36,112,308,518,588,428,181,34,1,45,156,
%U 504,1008,1428,1380,873,327,55,1,55,210,780,1806,3066,3690,3105
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208519; see the Formula section.
%C coefficient of x^(n-1): = Fibonacci(n) = A000045(n)
%C col 1: A000012
%C col 2: A000217 (triangular numbers)
%C col 3: A005581
%C col 4: A117662
%C alternating row sums: signed version of (-1+Fibonacci(n))
%F u(n,x)=u(n-1,x)+x*v(n-1,x),
%F v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%e First five rows:
%e 1
%e 1...1
%e 1...3....2
%e 1...6....7....3
%e 1...10...16...14...5
%e First five polynomials u(n,x):
%e 1
%e 1 + x
%e 1 + 3x + 2x^2
%e 1 + 6x + 7x^2 + 3x^3
%e 1 + 10x + 16x^2 + 14x^3 + 5x^4
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%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[%] (* A208518 *)
%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[%] (* A208519 *)
%Y Cf. A208519.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Feb 28 2012
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