A210798 Triangle of coefficients of polynomials v(n,x) jointly generated with A210797; see the Formula section.

1, 2, 2, 1, 3, 3, 2, 5, 7, 5, 1, 6, 12, 13, 8, 2, 8, 20, 29, 25, 13, 1, 9, 27, 51, 62, 46, 21, 2, 11, 39, 84, 125, 129, 84, 34, 1, 12, 48, 126, 224, 284, 258, 151, 55, 2, 14, 64, 182, 374, 562, 622, 505, 269, 89, 1, 15, 75, 250, 580, 1008, 1328, 1315, 969, 475

Offset: 1

Author

Clark Kimberling, Mar 26 2012

Keywords

• nonn
• tabl

Comments

Row n starts with A109613(n) and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Column 2: A114113

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

Examples

			First five rows:
1
2...2
1...3...3
2...5...7....5
1...6...12...13...8
First three polynomials v(n,x): 1, 2 + 2x, 1 + 3x + 3x^2
		

Crossrefs

Cf. A210797, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = 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[%]    (* A210797 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210798 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A099232 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A006130 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A008346 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A039834 *)

Formula

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

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

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