A210549 Triangle of coefficients of polynomials u(n,x) jointly generated with A210550; see the Formula section.

1, 1, 2, 1, 3, 4, 1, 3, 9, 8, 1, 3, 10, 25, 16, 1, 3, 10, 33, 65, 32, 1, 3, 10, 34, 104, 161, 64, 1, 3, 10, 34, 115, 311, 385, 128, 1, 3, 10, 34, 116, 381, 886, 897, 256, 1, 3, 10, 34, 116, 395, 1224, 2421, 2049, 512, 1, 3, 10, 34, 116, 396, 1334, 3796, 6388

Offset

1,3

Comments

Each row begins with 1 and ends with 2^(n-1).

Row sums: 1,3,8,21,..., the even-indexed Fibonacci numbers

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

Links

Table of n, a(n) for n=1..64.

Formula

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

v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,

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

Example

First five rows:
1
1...2
1...3...4
1...3...9....8
1...3...10...25...16
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 4x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%]      (* A210235 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]      (* A210236 *)

Crossrefs

Cf. A210550, A208510.

Sequence in context: A027422 A135086 A210561 * A187002 A177226 A059026

Adjacent sequences: A210546 A210547 A210548 * A210550 A210551 A210552

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 22 2012

Status

approved