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Triangle of coefficients of polynomials u(n,x) jointly generated with A210550; see the Formula section.
2

%I #6 Mar 30 2012 18:58:16

%S 1,1,2,1,3,4,1,3,9,8,1,3,10,25,16,1,3,10,33,65,32,1,3,10,34,104,161,

%T 64,1,3,10,34,115,311,385,128,1,3,10,34,116,381,886,897,256,1,3,10,34,

%U 116,395,1224,2421,2049,512,1,3,10,34,116,396,1334,3796,6388

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210550; see the Formula section.

%C Each row begins with 1 and ends with 2^(n-1).

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

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

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

%e 1...3...4

%e 1...3...9....8

%e 1...3...10...25...16

%e First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 4x^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 + 1)*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[%] (* A210235 *)

%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[%] (* A210236 *)

%Y Cf. A210550, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 22 2012