A208608 Triangle of coefficients of polynomials u(n,x) jointly generated with A208609; see the Formula section.

1, 1, 1, 1, 3, 2, 1, 5, 6, 3, 1, 7, 12, 12, 5, 1, 9, 20, 29, 23, 8, 1, 11, 30, 56, 64, 43, 13, 1, 13, 42, 95, 140, 136, 79, 21, 1, 15, 56, 148, 265, 332, 279, 143, 34, 1, 17, 72, 217, 455, 692, 751, 558, 256, 55, 1, 19, 90, 304, 728, 1295, 1708, 1641, 1093, 454

Offset: 1

Author

Clark Kimberling, Feb 29 2012

Keywords

• nonn
• tabl

Comments

coefficient of x^(n-1)=Fibonacci(n)=A000045(n)

Examples

			First five rows:
1
1...1
1...3...2
1...5...6....3
1...7...12...12...5
First five polynomials u(n,x):
1
1 + x
1 + 3x + 2x^2
1 + 5x + 6x^2 + 3x^3
1 + 7x + 12x^2 + 12x^3 + 5x^4
		

Crossrefs

Cf. A208609.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
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[%]  (* A208608 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208609 *)

Formula

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

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

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