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Triangle of coefficients of polynomials u(n,x) jointly generated with A210740; see the Formula section.
3

%I #6 Mar 30 2012 18:58:17

%S 1,1,3,1,4,8,1,4,14,21,1,4,15,46,55,1,4,15,55,145,144,1,4,15,56,196,

%T 444,377,1,4,15,56,208,678,1331,987,1,4,15,56,209,764,2282,3926,2584,

%U 1,4,15,56,209,779,2762,7499,11434,6765,1,4,15,56,209,780,2892

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210740; see the Formula section.

%C Rows end with even-indexed Fibonacci numbers

%C Limiting row: A001353

%C Row sums: A003562

%C Alternating row sums: A000975 (signed)

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

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

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

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

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

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

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

%e First five rows:

%e 1

%e 1...3

%e 1...4...8

%e 1...4...14...21

%e 1...4...15...46...55

%e First three polynomials u(n,x): 1, 1+ 3x, 1 + 4x + 8x^2.

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

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

%t v[n_, x_] := (x + 1)*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[%] (* A210739 *)

%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[%] (* A210740 *)

%Y Cf. A210740, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 24 2012