A210560 Triangle of coefficients of polynomials v(n,x) jointly generated with A210559; see the Formula section.

1, 3, 1, 5, 4, 2, 7, 9, 9, 3, 9, 16, 23, 16, 5, 11, 25, 46, 48, 30, 8, 13, 36, 80, 110, 101, 54, 13, 15, 49, 127, 215, 257, 203, 97, 21, 17, 64, 189, 378, 552, 570, 401, 172, 34, 19, 81, 268, 616, 1057, 1337, 1228, 776, 303, 55, 21, 100, 366, 948, 1862, 2772

Offset

1,2

Comments

Column 1: odd positive integers (A005408)

Column 2: squares (A000290)

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

Row sums: A005409

Alternating row sums: 1,2,3,4,5,6,7,8,...(A000027)

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

Links

Table of n, a(n) for n=1..61.

Formula

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

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
3...1
5...4...2
7...9...9...3
9...16...23...16...5
First three polynomials v(n,x): 1, 3 + x , 5 + 4x + 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 + 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[%]  (* A210559 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210560 *)

Crossrefs

Cf. A210559, A208510.

Sequence in context: A308676 A131809 A016574 * A208922 A209770 A210799

Adjacent sequences: A210557 A210558 A210559 * A210561 A210562 A210563

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 22 2012

Status

approved