A208609 Triangle of coefficients of polynomials v(n,x) jointly generated with A208608; see the Formula section.

1, 2, 2, 2, 4, 3, 2, 6, 9, 5, 2, 8, 17, 18, 8, 2, 10, 27, 41, 35, 13, 2, 12, 39, 76, 93, 66, 21, 2, 14, 53, 125, 196, 200, 122, 34, 2, 16, 69, 190, 360, 472, 415, 222, 55, 2, 18, 87, 273, 603, 957, 1083, 837, 399, 89, 2, 20, 107, 376, 945, 1750, 2400, 2392

Offset

1,2

Comments

v(n,n)=Fibonacci(n+1)=A000045(n+1).

Links

Table of n, a(n) for n=1..63.

Formula

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

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

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

Example

First five rows:
1
2...2
2...4...3
2...6...9....5
2...8...17...18...8
First five polynomials v(n,x):
1
2 + 2x
2 + 4x + 3x^2
2 + 6x + 9x^2 + 5x^3
2 + 8x + 17x^2 + 18x^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 + 1)*u[n - 1, x] + 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[%]  (* A208608 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208609 *)

Crossrefs

Cf. A208608.

Sequence in context: A283681 A222819 A194319 * A249030 A358527 A257126

Adjacent sequences: A208606 A208607 A208608 * A208610 A208611 A208612

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 29 2012

Status

approved