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

1, 1, 1, 1, 3, 2, 1, 4, 5, 3, 1, 6, 10, 10, 5, 1, 7, 16, 22, 18, 8, 1, 9, 24, 42, 47, 33, 13, 1, 10, 33, 69, 98, 95, 59, 21, 1, 12, 44, 108, 182, 220, 188, 105, 34, 1, 13, 56, 156, 308, 444, 472, 363, 185, 55, 1, 15, 70, 220, 490, 818, 1034, 985, 690, 324, 89, 1

Offset

1,5

Comments

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

Column 2: A032766

Column 3: A001859

Row sums: A099232

Alternating row sums: A008346

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

Links

Table of n, a(n) for n=1..67.

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.

Example

First five rows:
1
1...1
1...3...2
1...4...5....3
1...6...10...10...5
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + 2x^2.

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 *)

Crossrefs

Cf. A210798, A208510.

Sequence in context: A336886 A261876 A272336 * A222220 A271830 A193815

Adjacent sequences: A210794 A210795 A210796 * A210798 A210799 A210800

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 26 2012

Status

approved