|
|
A210873
|
|
Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.
|
|
4
|
|
|
1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 1, 2, 8, 5, 1, 1, 2, 6, 17, 6, 1, 1, 2, 5, 18, 31, 7, 1, 1, 2, 5, 14, 47, 51, 8, 1, 1, 2, 5, 13, 41, 107, 78, 9, 1, 1, 2, 5, 13, 35, 115, 218, 113, 10, 1, 1, 2, 5, 13, 34, 98, 296, 407, 157, 11, 1, 1, 2, 5, 13, 34, 90, 276, 695, 709, 211, 12
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Column 1: 1,1,1,1,1,1,1,1,1...
Limiting row: 1,1,2,5,13,34,..., odd-indexed Fibonacci numbers
If the term in row n and column k is written as U(n,k), then U(n,n-1)=A105163.
For a discussion and guide to related arrays, see A208510.
|
|
LINKS
|
|
|
FORMULA
|
For a discussion and guide to related arrays, see A208510.
u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
|
|
EXAMPLE
|
First six rows:
1
1...2
1...1...3
1...1...3....4
1...1...2....8...5
1...1...2....6...17...6
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2
|
|
MATHEMATICA
|
u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
v[n_, x_] := x*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]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A083318 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* -A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|