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Triangle of coefficients of polynomials u(n,x) jointly generated with A210218; see the Formula section.
4

%I #4 Mar 30 2012 18:58:16

%S 1,2,1,2,5,1,2,6,12,1,2,6,19,27,1,2,6,20,57,58,1,2,6,20,67,160,121,1,

%T 2,6,20,68,218,424,248,1,2,6,20,68,231,680,1073,503,1,2,6,20,68,232,

%U 775,2028,2619,1014,1,2,6,20,68,232,791,2543,5797,6214,2037,1,2

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210218; see the Formula section.

%C Limiting row: A006012

%C Row sums: even-indexed Fibonacci numbers: 1,3,8,21,...

%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)=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 2...5...1

%e 2...6...12...1

%e 2...6...19...27...1

%e First three polynomials u(n,x): 1, 2 + x, 2 + 5x + 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_] := 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[%] (* A210217 *)

%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[%] (* A210218 *)

%Y Cf. A210218, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 19 2012