A209167 Triangle of coefficients of polynomials v(n,x) jointly generated with A209166; see the Formula section.

1, 2, 2, 3, 5, 3, 5, 12, 12, 5, 8, 25, 35, 25, 8, 13, 50, 89, 89, 50, 13, 21, 96, 207, 263, 207, 96, 21, 34, 180, 455, 698, 698, 455, 180, 34, 55, 331, 959, 1719, 2073, 1719, 959, 331, 55, 89, 600, 1959, 4011, 5643, 5643, 4011, 1959, 600, 89, 144, 1075

Offset

1,2

Comments

Row n begins and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,1,0,...

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

Links

Table of n, a(n) for n=1..57.

Formula

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

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

where u(1,x)=1, v(1,x)=1.

Example

First five rows:
  1
  2    2
  3    5    3
  5    12   12   5
  8    25   35   25   8
First three polynomials v(n,x): 1, 2 + 2*x, 3 + 5*x + 3*x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209166 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209167 *)

Crossrefs

Cf. A209166, A208510.

Sequence in context: A317043 A317697 A132403 * A299995 A113167 A036014

Adjacent sequences: A209164 A209165 A209166 * A209168 A209169 A209170

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 08 2012

Status

approved