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Triangle of coefficients of polynomials v(n,x) jointly generated with A209755; see the Formula section.
3

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

%S 1,2,1,3,2,2,4,4,6,3,5,7,13,11,5,6,11,24,28,22,8,7,16,40,59,63,41,13,

%T 8,22,62,110,146,132,76,21,9,29,91,188,296,337,271,138,34,10,37,128,

%U 301,546,743,754,541,248,55,11,46,174,458,938,1477,1793,1632

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209755; see the Formula section.

%C Column 1: 1,2,3,4,5,6,....... A000027

%C Column 2: 1,2,4,7,11,........ A000124

%C Column 3: 2,6,13,24,......... A003600

%C Last term in row n: 1,1,2,3,. A000045 (Fibonacci numbers)

%C Row sums: 1,3,7,17,41,...... A001333

%C Alternating row sums: 1,2,3,3,5,5,7,7,...; A109613

%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*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...1

%e 3...2...2

%e 4...4...6....3

%e 5...7...13...11...5

%e First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 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*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[%] (* A209755 *)

%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[%] (* A209756 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A133654 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A001333 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A109613 *)

%Y Cf. A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 14 2012