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Triangle of coefficients of polynomials v(n,x) jointly generated with A208518; see the Formula section.
3

%I #9 Mar 30 2012 18:58:13

%S 1,2,2,3,5,3,4,9,11,5,5,14,26,23,8,6,20,50,65,45,13,7,27,85,145,150,

%T 86,21,8,35,133,280,385,329,160,34,9,44,196,490,840,952,692,293,55,10,

%U 54,276,798,1638,2310,2232,1413,529,89,11,65,375,1230,2940,4956

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208518; see the Formula section.

%C coefficient of x^(n-1): Fibonacci(n+1) = A000045(n+1)

%C col 1: A000027

%C col 2: A000096

%C col 3: A051925

%C row sums: A002878 (bisection of Lucas sequence)

%C alternating row sums: A000045(n-2), Fibonacci numbers

%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 2...2

%e 3...5....3

%e 4...9....11...5

%e 5...14...26...23...8

%e First five polynomials v(n,x):

%e 1

%e 2 + 2x

%e 3 + 5x + 3x^2

%e 4 + 9x + 11x^2 + 5x^3

%e 5 + 14x + 26x^2 + 23x^3 + 8x^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. A208518.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 28 2012