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Triangle of coefficients of polynomials v(n,x) jointly generated with A210797; see the Formula section.
3

%I #5 Mar 30 2012 18:58:17

%S 1,2,2,1,3,3,2,5,7,5,1,6,12,13,8,2,8,20,29,25,13,1,9,27,51,62,46,21,2,

%T 11,39,84,125,129,84,34,1,12,48,126,224,284,258,151,55,2,14,64,182,

%U 374,562,622,505,269,89,1,15,75,250,580,1008,1328,1315,969,475

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210797; see the Formula section.

%C Row n starts with A109613(n) and ends with F(n+1), where F=A000045 (Fibonacci numbers).

%C Column 2: A114113

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

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

%F v(n,x)=(x+2)*u(n-1,x)+(x-1)*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 1...3...3

%e 2...5...7....5

%e 1...6...12...13...8

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

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

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

%t d[x_] := h + x; e[x_] := p + x;

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

%t j = 0; c = 0; h = 2; p = -1; f = 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[%] (* A210797 *)

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A210798 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A099232 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A006130 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A008346 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A039834 *)

%Y Cf. A210797, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 26 2012