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

1,5

Comments

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

Links

Table of n, a(n) for n=1..65.

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.

Example

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

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 *)

Crossrefs

Cf. A208609.

Sequence in context: A132969 A132970 A192022 * A209577 A139377 A368607

Adjacent sequences: A208605 A208606 A208607 * A208609 A208610 A208611

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 29 2012

Status

approved