A209766 Triangle of coefficients of polynomials v(n,x) jointly generated with A209765; see the Formula section.

1, 1, 3, 1, 3, 7, 1, 3, 13, 17, 1, 3, 13, 43, 41, 1, 3, 13, 55, 133, 99, 1, 3, 13, 55, 209, 391, 239, 1, 3, 13, 55, 233, 739, 1113, 577, 1, 3, 13, 55, 233, 939, 2469, 3095, 1393, 1, 3, 13, 55, 233, 987, 3589, 7903, 8457, 3363, 1, 3, 13, 55, 233, 987, 4085

Offset

1,3

Comments

Limiting row: F(1+3k), where F=A000045 (Fibonacci numbers)

Coefficient of x^n in u(n,x): A001333(n)

Row sums: 1,4,11,34,101,304,... A060925.

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

Links

Table of n, a(n) for n=1..62.

Formula

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

v(n,x)=2x*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...3
1...3...7
1...3...13...17
1...3...13...43...41
First three polynomials v(n,x): 1, 1 + 3x , 1 + 3x + 7x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 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]
Flatten[%]    (* A209765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209766 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A081250 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A060925 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A042963 signed *)

Crossrefs

Cf. A209665, A208510.

Sequence in context: A336203 A209566 A208916 * A356207 A114972 A107461

Adjacent sequences: A209763 A209764 A209765 * A209767 A209768 A209769

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 14 2012

Status

approved