A208519 Triangle of coefficients of polynomials v(n,x) jointly generated with A208518; see the Formula section.

1, 2, 2, 3, 5, 3, 4, 9, 11, 5, 5, 14, 26, 23, 8, 6, 20, 50, 65, 45, 13, 7, 27, 85, 145, 150, 86, 21, 8, 35, 133, 280, 385, 329, 160, 34, 9, 44, 196, 490, 840, 952, 692, 293, 55, 10, 54, 276, 798, 1638, 2310, 2232, 1413, 529, 89, 11, 65, 375, 1230, 2940, 4956

Offset

1,2

Comments

coefficient of x^(n-1): Fibonacci(n+1) = A000045(n+1)

col 1: A000027

col 2: A000096

col 3: A051925

row sums: A002878 (bisection of Lucas sequence)

alternating row sums: A000045(n-2), Fibonacci numbers

Links

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

Formula

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

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

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

Example

First five rows:
1
2...2
3...5....3
4...9....11...5
5...14...26...23...8
First five polynomials v(n,x):
1
2 + 2x
3 + 5x + 3x^2
4 + 9x + 11x^2 + 5x^3
5 + 14x + 26x^2 + 23x^3 + 8x^4

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*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[%]   (* A208518 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208519 *)

Crossrefs

Cf. A208518.

Sequence in context: A124727 A210565 A125101 * A336725 A210232 A047666

Adjacent sequences: A208516 A208517 A208518 * A208520 A208521 A208522

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 28 2012

Status

approved