A210870 Triangle of coefficients of polynomials u(n,x) jointly generated with A210871; see the Formula section.

1, 1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 4, 6, 5, 5, 1, 5, 8, 12, 8, 8, 1, 6, 12, 17, 23, 13, 13, 1, 7, 15, 29, 33, 43, 21, 21, 1, 8, 20, 38, 64, 63, 79, 34, 34, 1, 9, 24, 56, 86, 136, 117, 143, 55, 55, 1, 10, 30, 70, 140, 187, 279, 214, 256, 89, 89, 1, 11, 35, 95, 180, 332

Offset: 1

Author

Clark Kimberling, Mar 29 2012

Keywords

• nonn
• tabl

Comments

In row n, for n>1, the first two terms are 1 and n-1, and the last two are F(n) and F(n), where F = A000045 (Fibonacci numbers).

Row sums: A000975

Alternating row sums: A113954

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

Examples

			First six rows:
1
1...1
1...2...2
1...3...3...3
1...4...6...5....5
1...5...8...12...8...8
First three polynomials u(n,x): 1, 1 + x, 1 + 2x + 2x^2.
		

Crossrefs

Cf. A210871, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
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[%]    (* A210870 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210871 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000975 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A001045 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]   (* A113954 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A077925 *)

Formula

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

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.