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Triangle of coefficients of polynomials v(n,x) jointly generated with A209569; see the Formula section.
3

%I #8 Mar 30 2012 18:58:15

%S 1,2,2,3,4,2,4,8,8,2,5,14,20,12,2,6,22,42,40,16,2,7,32,78,102,68,20,2,

%T 8,44,132,222,210,104,24,2,9,58,208,432,534,382,148,28,2,10,74,310,

%U 772,1188,1126,634,200,32,2,11,92,442,1290,2392,2848,2142,982

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209569; see the Formula section.

%C For n>1, row n begins and ends with 2.

%C Alternating row sums: 1,0,1,2,1,0,1,2,1,0,1,2,...

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

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

%F v(n,x)=2x*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 2...2

%e 3...4....2

%e 4...8....8....2

%e 5...14...20...12...2

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

%t v[n_, x_] := 2 x*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[%] (* A209569 *)

%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[%] (* A209570 *)

%Y Cf. A209569, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 10 2012