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A210216 Triangle of coefficients of polynomials v(n,x) jointly generated with A210215; see the Formula section. 3

%I

%S 1,1,2,1,3,3,1,3,7,4,1,3,8,14,5,1,3,8,20,25,6,1,3,8,21,46,41,7,1,3,8,

%T 21,54,97,63,8,1,3,8,21,55,133,189,92,9,1,3,8,21,55,143,309,344,129,

%U 10,1,3,8,21,55,144,364,674,591,175,11,1,3,8,21,55,144,376,894

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210215; see the Formula section.

%C Limiting row: even-indexed Fibonacci numbers, A001906.

%C n-th row sum: -1+2*n

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

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

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

%e 1...3...7...4

%e 1...3...8...14...5

%e First three polynomials v(n,x): 1, 1 + 2x , 1 + 3x + 3x^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*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[%] (* A210215 *)

%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[%] (* A210216 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

%Y Cf. A210215, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 19 2012

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Last modified April 3 23:48 EDT 2020. Contains 333207 sequences. (Running on oeis4.)