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

Author

Clark Kimberling, Feb 28 2012

Keywords

• nonn
• tabl

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

Examples

			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
		

Crossrefs

Cf. A208518.

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.