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

1, 2, 1, 3, 2, 2, 4, 4, 6, 3, 5, 7, 13, 11, 5, 6, 11, 24, 28, 22, 8, 7, 16, 40, 59, 63, 41, 13, 8, 22, 62, 110, 146, 132, 76, 21, 9, 29, 91, 188, 296, 337, 271, 138, 34, 10, 37, 128, 301, 546, 743, 754, 541, 248, 55, 11, 46, 174, 458, 938, 1477, 1793, 1632

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,......... A003600

Last term in row n: 1,1,2,3,. A000045 (Fibonacci numbers)

Row sums: 1,3,7,17,41,...... A001333

Alternating row sums: 1,2,3,3,5,5,7,7,...; A109613

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

Examples

			First five rows:
1
2...1
3...2...2
4...4...6....3
5...7...13...11...5
First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.
		

Crossrefs

Cf. 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.