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

Author

Clark Kimberling, Feb 28 2012

Keywords

• nonn
• tabl

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))

Examples

			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
		

Crossrefs

Cf. A208519.

Programs

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 *)

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.