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Triangle of coefficients of polynomials u(n,x) jointly generated with A210212; see the Formula section.
3

%I #7 Jul 12 2012 00:40:00

%S 1,2,1,3,4,1,4,8,8,1,5,14,19,16,1,6,21,42,42,32,1,7,30,72,114,89,64,1,

%T 8,40,120,216,290,184,128,1,9,52,178,414,593,706,375,256,1,10,65,260,

%U 670,1292,1531,1666,758,512,1,11,80,355,1090,2247,3754,3782

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210212; see the Formula section.

%C Row n starts with n and ends with 2^n followed by 1.

%C n-th row sum: F(2k), where F=A000045 (Fibonacci numbers)

%C Alternating row sums are signed products of two Fibonacci numbers.

%C For a discussion and guide to related arrays, see A208510.

%F u(n,x)=x*u(n-1,x)+v(n-1,x)+1,

%F v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 2...1

%e 3...4....1

%e 4...8....8....1

%e 5...14...19...16...1

%e First three polynomials u(n,x): 1, 2 + x, 3 + 4x + x^2.

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;

%t v[n_, x_] := u[n - 1, x] + 2 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[%] (* A210211 *)

%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[%] (* A210212 *)

%Y Cf. A210204, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 19 2012