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Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.
3

%I #5 Mar 30 2012 18:58:16

%S 1,2,1,3,2,2,4,3,5,3,5,4,9,8,5,6,5,14,15,15,8,7,6,20,24,31,26,13,8,7,

%T 27,35,54,57,46,21,9,8,35,48,85,104,108,80,34,10,9,44,63,125,170,209,

%U 199,139,55,11,10,54,80,175,258,360,404,366,240,89,12,11,65,99

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.

%C Let T(n,k) denote the term in row n, column k.

%C T(n,n): A000045 (Fibonacci numbers)

%C T(n,n-1): A006367

%C T(n,n-2): A105423

%C T(n,1): 1,2,3,4,5,6,7,8,9,...

%C T(n,2): 1,2,3,4,5,6,7,8,9,...

%C T(n,3): A000096

%C T(n,4): A005563

%C T(n,5): A055831

%C T(n,6): A111694

%C Row sums: A000225

%C Alternating row sums: A052551

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

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

%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...3...5...3

%e 5...4...9...8...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*v[n - 1, x] + 1;

%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[%] (* A210552 *)

%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[%] (* A210553 *)

%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}] (* A094024 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

%Y Cf. A210552, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 22 2012