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Triangle of coefficients of polynomials v(n,x) jointly generated with A209769; see the Formula section.
3

%I #5 Mar 30 2012 18:58:15

%S 1,3,1,5,4,2,9,12,10,3,15,29,33,19,5,25,64,93,77,37,8,41,132,234,251,

%T 171,69,13,67,261,548,719,629,362,127,21,109,500,1216,1884,2004,1482,

%U 742,230,34,177,936,2592,4628,5784,5196,3342,1482,412,55,287

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209769; see the Formula section.

%C Column 1: A001595

%C Row n ends with F(n), where F=A000045, the Fibonacci numbers.

%C Row sums: 1,4,11,34,101,304,911,2734,... A060925

%C Alternating row sums: 1,2,3,4,5,6,7,.... A000027

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

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

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

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

%e First five rows:

%e 1

%e 3....1

%e 5....4....2

%e 9....12...10...3

%e 15...29...33...19...5

%e First three polynomials v(n,x): 1, 3 + x , 5 + 4x + 2x^2.

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

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

%t v[n_, x_] := (x + 1)*u[n - 1, 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[%] (* A209769 *)

%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[%] (* A209770 *)

%Y Cf. A209669, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 15 2012