A210232 Triangle of coefficients of polynomials v(n,x) jointly generated with A210231; see the Formula section.

1, 2, 2, 3, 5, 3, 4, 10, 10, 4, 5, 16, 25, 18, 5, 6, 24, 48, 54, 30, 6, 7, 33, 84, 123, 106, 47, 7, 8, 44, 132, 246, 282, 194, 70, 8, 9, 56, 198, 438, 637, 594, 336, 100, 9, 10, 70, 280, 730, 1272, 1504, 1170, 556, 138, 10, 11, 85, 385, 1140, 2337, 3337, 3301

Offset: 1

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn
• tabl

Comments

First and last terms of row n are n.

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,...

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

Examples

			First five rows:
1
2...2
3...5....3
4...10...10...4
5...16...25...16...5
First three polynomials v(n,x): 1, 2 + 2x , 3 + 5x + 3x^2.
		

Crossrefs

Cf. A210231, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + 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[%]     (* A210231 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210232 *)

Formula

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

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

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