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

1, 1, 1, 1, 3, 2, 1, 6, 7, 3, 1, 10, 16, 14, 5, 1, 15, 30, 40, 28, 8, 1, 21, 50, 90, 93, 53, 13, 1, 28, 77, 175, 238, 203, 99, 21, 1, 36, 112, 308, 518, 588, 428, 181, 34, 1, 45, 156, 504, 1008, 1428, 1380, 873, 327, 55, 1, 55, 210, 780, 1806, 3066, 3690, 3105

Offset

1,5

Comments

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

col 1: A000012

col 2: A000217 (triangular numbers)

col 3: A005581

col 4: A117662

alternating row sums: signed version of (-1+Fibonacci(n))

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*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
1...1
1...3....2
1...6....7....3
1...10...16...14...5
First five polynomials u(n,x):
1
1 + x
1 + 3x + 2x^2
1 + 6x + 7x^2 + 3x^3
1 + 10x + 16x^2 + 14x^3 + 5x^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. A208519.

Sequence in context: A079513 A060408 A267121 * A139624 A132276 A257558

Adjacent sequences: A208515 A208516 A208517 * A208519 A208520 A208521

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 28 2012

Status

approved