A209157 Triangle of coefficients of polynomials v(n,x) jointly generated with A209154; see the Formula section.

1, 2, 2, 3, 6, 2, 4, 14, 12, 4, 5, 28, 40, 24, 4, 6, 50, 104, 96, 40, 8, 7, 82, 234, 304, 204, 72, 8, 8, 126, 476, 820, 768, 408, 112, 16, 9, 184, 896, 1968, 2408, 1760, 768, 192, 16, 10, 258, 1584, 4320, 6640, 6288, 3776, 1408, 288, 32, 11, 350, 2658

Offset: 1

Author

Clark Kimberling, Mar 07 2012

Keywords

• nonn
• tabl

Comments

Last number of each row is a power of 2.

(n-th alternating row sum)=2-n for n>1.

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

Examples

			First five rows:
1
2...2
3...6....2
4...14...12...4
5...28...40...24...4
First three polynomials v(n,x): 1, 2 + 2x, 3 + 6x + x^2.
		

Crossrefs

Cf. A209154, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 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[%]    (* A209154 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209157 *)

Formula

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

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

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