A209769 Triangle of coefficients of polynomials u(n,x) jointly generated with A209770; see the Formula section.

1, 1, 2, 3, 5, 3, 5, 12, 11, 5, 9, 26, 34, 24, 8, 15, 53, 88, 86, 48, 13, 25, 104, 210, 258, 200, 93, 21, 41, 198, 470, 695, 680, 440, 175, 34, 67, 369, 1007, 1737, 2043, 1671, 929, 323, 55, 109, 676, 2085, 4107, 5625, 5529, 3895, 1901, 587, 89, 177

Offset

1,3

Comments

Column 1: A001595

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

Row sums: 1,3,11,33,101,303,911,2733,... A081250

Alternating row sums: 1,-1,1,-1,1,-1,... A033999

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

Links

Table of n, a(n) for n=1..56.

Formula

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

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

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

Example

First five rows:
1
1...2
3...5....3
5...12...11...5
9...26...34...24...8
First three polynomials u(n,x): 1, 1 + 2x, 3 + 5x + 3x^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_] := (x + 1)*u[n - 1, 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[%]    (* A209769 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209770 *)

Crossrefs

Cf. A209770, A208510.

Sequence in context: A151571 A193957 A336746 * A114230 A209753 A185191

Adjacent sequences: A209766 A209767 A209768 * A209770 A209771 A209772

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 15 2012

Status

approved