A209755 Triangle of coefficients of polynomials u(n,x) jointly generated with A209756; see the Formula section.

1, 1, 2, 2, 4, 3, 3, 7, 8, 5, 4, 11, 17, 17, 8, 5, 16, 31, 41, 33, 13, 6, 22, 51, 83, 91, 63, 21, 7, 29, 78, 150, 205, 195, 117, 34, 8, 37, 113, 250, 406, 483, 403, 214, 55, 9, 46, 157, 392, 734, 1039, 1091, 812, 386, 89, 10, 56, 211, 586, 1239, 2023, 2536

Offset: 1

Author

Clark Kimberling, Mar 14 2012

Keywords

• nonn
• tabl

Comments

Column 1: 1,2,3,4,5,6,....... A000027

Column 2: 1,2,4,7,11,........ A000124

Column 3: 2,6,13,24,......... A105163

Final row terms: 1,2,3,5,.... A000045 (Fibonacci numbers)

Row sums: 1,3,9,23,57,139,... A133654

Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...; A033999

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

Examples

			First five rows:
1
1...2
2...4....3
3...7....8....5
4...11...11...17...8
First three polynomials u(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.
		

Crossrefs

Cf. A209756, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, 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[%]    (* A209755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209756 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A133654 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A001333 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A109613 *)

Formula

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

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

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