A210747 Triangle of coefficients of polynomials u(n,x) jointly generated with A210748; see the Formula section.

1, 2, 3, 4, 9, 8, 7, 24, 33, 21, 12, 54, 109, 111, 55, 20, 114, 297, 435, 355, 144, 33, 228, 736, 1383, 1606, 1098, 377, 54, 441, 1697, 3912, 5813, 5625, 3316, 987, 88, 831, 3723, 10158, 18419, 22779, 18962, 9837, 2584, 143, 1536, 7859, 24798

Offset

1,2

Comments

Row n starts with -1+F(n+2) and ends with F(2n), where F=A000045 (Fibonacci numbers).

Row sums: A002450

Alternating row sums: A077925

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

Links

Table of n, a(n) for n=1..49.

Formula

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

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

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

Example

First five rows:
1
2....3
4....9....8
7....24...33....21
12...54...109...111...55
First three polynomials u(n,x): 1, 2+ 3x, 4 + 9x + 8x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%]    (* A210747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210748 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A002450 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)

Crossrefs

Cf. A208510.

Sequence in context: A307404 A307405 A115305 * A353239 A351497 A329425

Adjacent sequences: A210744 A210745 A210746 * A210748 A210749 A210750

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 25 2012

Status

approved