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%I #8 Dec 20 2021 06:50:23
%S 1,1,1,1,2,2,1,3,4,3,1,4,6,7,5,1,5,8,12,13,8,1,6,10,18,24,23,13,1,7,
%T 12,25,38,46,41,21,1,8,14,33,55,78,88,72,34,1,9,16,42,75,120,158,165,
%U 126,55,1,10,18,52,98,173,255,313,307,219,89,1,11,20,63,124,238
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208515; see the Formula section.
%C u(n,n) = Fibonacci(n), A000045
%C u(n+1,n) = A208354(n)
%C col 1: A000012
%C col 2: A000027
%C col 3: A005843
%C col 4: A055998
%C col 5: A140090
%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)+1,
%F where u(1,x)=1, v(1,x)=1.
%e First five rows:
%e 1
%e 1...1
%e 1...2...2
%e 1...3...4...3
%e 1...4...6...7...5
%e First five polynomials u(n,x):
%e 1
%e 1 + x
%e 1 + 2x + 2x^2
%e 1 + 3x + 4x^2 + 3x^3
%e 1 + 4x + 6x^2 + 7x^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*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[%] (* A208514 *)
%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[%] (* A208515 *)
%Y Cf. A208515.
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_, Feb 28 2012
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