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

Author

Clark Kimberling, Mar 25 2012

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Crossrefs

Cf. A208510.

Programs

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 *)

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.