A210204 Triangle of coefficients of polynomials v(n,x) jointly generated with A210203; see the Formula section.

1, 3, 2, 7, 8, 2, 15, 24, 12, 2, 31, 64, 48, 16, 2, 63, 160, 160, 80, 20, 2, 127, 384, 480, 320, 120, 24, 2, 255, 896, 1344, 1120, 560, 168, 28, 2, 511, 2048, 3584, 3584, 2240, 896, 224, 32, 2, 1023, 4608, 9216, 10752, 8064, 4032, 1344, 288, 36, 2, 2047

Offset: 1

Author

Clark Kimberling, Mar 18 2012

Keywords

• nonn
• tabl

Comments

Column 1: -1+2^n.

Row sums: A048473.

Alternating row sums: 1,1,1,1,1,1,1,1,1,...

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

Row sums without first column give A056182. - Alois P. Heinz, Jan 14 2022

Examples

			First five rows:
1
3....2
7....8....2
15...24...12...2
31...64...48...16...2
First three polynomials v(n,x): 1, 3 + 2x , 7 + 8x + 2x^2.
		

Crossrefs

Cf. A210203, A208510.

Cf. A056182.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 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[%]   (* A210203 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210204 *)

Formula

u(n,x)=u(n-1,x)+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.