A210187 Triangle of coefficients of polynomials u(n,x) jointly generated with A210188; see the Formula section.

1, 3, 6, 2, 10, 9, 2, 15, 25, 13, 2, 21, 55, 49, 17, 2, 28, 105, 140, 81, 21, 2, 36, 182, 336, 285, 121, 25, 2, 45, 294, 714, 825, 506, 169, 29, 2, 55, 450, 1386, 2079, 1716, 819, 225, 33, 2, 66, 660, 2508, 4719, 5005, 3185, 1240, 289, 37, 2, 78, 935, 4290

Offset: 1

Author

Clark Kimberling, Mar 18 2012

Keywords

• nonn
• tabf

Comments

Row sums: even-indexed Fibonacci numbers: 1,3,8,21,55,...

Column 1: triangular numbers: A000217

Alternating row sums: periodic with period (1,3,4,3,1,0)

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

Examples

			First five rows:
1
3
6....2
10...9....2
15...25...13...2
First three polynomials u(n,x): 1, 3, 6 + 2x.
		

Crossrefs

Cf. A210188, A208510.

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*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[%]    (* A210187 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210188 *)

Formula

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

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

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