A210211 Triangle of coefficients of polynomials u(n,x) jointly generated with A210212; see the Formula section.

1, 2, 1, 3, 4, 1, 4, 8, 8, 1, 5, 14, 19, 16, 1, 6, 21, 42, 42, 32, 1, 7, 30, 72, 114, 89, 64, 1, 8, 40, 120, 216, 290, 184, 128, 1, 9, 52, 178, 414, 593, 706, 375, 256, 1, 10, 65, 260, 670, 1292, 1531, 1666, 758, 512, 1, 11, 80, 355, 1090, 2247, 3754, 3782

Offset

1,2

Comments

Row n starts with n and ends with 2^n followed by 1.

n-th row sum: F(2k), where F=A000045 (Fibonacci numbers)

Alternating row sums are signed products of two Fibonacci numbers.

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

Links

Table of n, a(n) for n=1..62.

Formula

u(n,x)=x*u(n-1,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
2...1
3...4....1
4...8....8....1
5...14...19...16...1
First three polynomials u(n,x): 1, 2 + x, 3 + 4x + x^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_] := u[n - 1, x] + 2 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[%]    (* A210211 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210212 *)

Crossrefs

Cf. A210204, A208510.

Sequence in context: A180378 A208341 A201634 * A283054 A247358 A297224

Adjacent sequences: A210208 A210209 A210210 * A210212 A210213 A210214

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 19 2012

Status

approved