A210553 Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.

1, 2, 1, 3, 2, 2, 4, 3, 5, 3, 5, 4, 9, 8, 5, 6, 5, 14, 15, 15, 8, 7, 6, 20, 24, 31, 26, 13, 8, 7, 27, 35, 54, 57, 46, 21, 9, 8, 35, 48, 85, 104, 108, 80, 34, 10, 9, 44, 63, 125, 170, 209, 199, 139, 55, 11, 10, 54, 80, 175, 258, 360, 404, 366, 240, 89, 12, 11, 65, 99

Offset

1,2

Comments

Let T(n,k) denote the term in row n, column k.

T(n,n): A000045 (Fibonacci numbers)

T(n,n-1): A006367

T(n,n-2): A105423

T(n,1): 1,2,3,4,5,6,7,8,9,...

T(n,2): 1,2,3,4,5,6,7,8,9,...

T(n,3): A000096

T(n,4): A005563

T(n,5): A055831

T(n,6): A111694

Row sums: A000225

Alternating row sums: A052551

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

Links

Table of n, a(n) for n=1..70.

Formula

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

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
2...1
3...2...2
4...3...5...3
5...4...9...8...5
First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
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[%]   (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

Crossrefs

Cf. A210552, A208510.

Sequence in context: A141157 A357213 A137948 * A269456 A208906 A120933

Adjacent sequences: A210550 A210551 A210552 * A210554 A210555 A210556

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 22 2012

Status

approved