A210876 Triangle of coefficients of polynomials u(n,x) jointly generated with A210877; see the Formula section.

1, 2, 1, 1, 5, 1, 1, 4, 9, 1, 1, 3, 12, 14, 1, 1, 3, 9, 29, 20, 1, 1, 3, 8, 27, 60, 27, 1, 1, 3, 8, 22, 74, 111, 35, 1, 1, 3, 8, 21, 63, 181, 189, 44, 1, 1, 3, 8, 21, 56, 178, 399, 302, 54, 1, 1, 3, 8, 21, 55, 154, 474, 806, 459, 65, 1, 1, 3, 8, 21, 55, 145, 430, 1169

Offset

1,2

Comments

For n>2, each row begins with 1 and ends with 1. If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A000096(n-1) and U(n,n-3)=A086274(n-1).

Row sums: A000225 (-1+2^n)

Alternating row sums: A077973

Limiting row: 1,3,8,21,55,..., even-indexed Fibonacci numbers

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

Links

Table of n, a(n) for n=1..74.

Formula

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

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

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

Example

First six rows:
1
2...1
1...5...1
1...4...9....1
1...3...12...14...1
1...3...9....29...20...1
First three polynomials u(n,x): 1, 2 + x, 1 + 5x + x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + x;
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[%]    (* A210876 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210877 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

Crossrefs

Cf. A210877, A208510.

Sequence in context: A069568 A210545 A141323 * A174785 A356399 A373173

Adjacent sequences: A210873 A210874 A210875 * A210877 A210878 A210879

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 30 2012

Status

approved