A209755 Triangle of coefficients of polynomials u(n,x) jointly generated with A209756; see the Formula section.

1, 1, 2, 2, 4, 3, 3, 7, 8, 5, 4, 11, 17, 17, 8, 5, 16, 31, 41, 33, 13, 6, 22, 51, 83, 91, 63, 21, 7, 29, 78, 150, 205, 195, 117, 34, 8, 37, 113, 250, 406, 483, 403, 214, 55, 9, 46, 157, 392, 734, 1039, 1091, 812, 386, 89, 10, 56, 211, 586, 1239, 2023, 2536

Offset

1,3

Comments

Column 1: 1,2,3,4,5,6,....... A000027

Column 2: 1,2,4,7,11,........ A000124

Column 3: 2,6,13,24,......... A105163

Final row terms: 1,2,3,5,.... A000045 (Fibonacci numbers)

Row sums: 1,3,9,23,57,139,... A133654

Alternating row sums: 1,-1,1,-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..62.

Formula

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

v(n,x)=x*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
2...4....3
3...7....8....5
4...11...11...17...8
First three polynomials u(n,x): 1, 1 + 2x, 2 + 4x + 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*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[%]    (* A209755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209756 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A133654 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A001333 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A109613 *)

Crossrefs

Cf. A209756, A208510.

Sequence in context: A143228 A143211 A361757 * A131052 A209138 A368149

Adjacent sequences: A209752 A209753 A209754 * A209756 A209757 A209758

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 14 2012

Status

approved