A208514 Triangle of coefficients of polynomials u(n,x) jointly generated with A208515; see the Formula section.

1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 6, 7, 5, 1, 5, 8, 12, 13, 8, 1, 6, 10, 18, 24, 23, 13, 1, 7, 12, 25, 38, 46, 41, 21, 1, 8, 14, 33, 55, 78, 88, 72, 34, 1, 9, 16, 42, 75, 120, 158, 165, 126, 55, 1, 10, 18, 52, 98, 173, 255, 313, 307, 219, 89, 1, 11, 20, 63, 124, 238

Offset: 1

Author

Clark Kimberling, Feb 28 2012

Keywords

• nonn
• tabl

Comments

u(n,n) = Fibonacci(n), A000045

u(n+1,n) = A208354(n)

col 1: A000012

col 2: A000027

col 3: A005843

col 4: A055998

col 5: A140090

Examples

			First five rows:
1
1...1
1...2...2
1...3...4...3
1...4...6...7...5
First five polynomials u(n,x):
1
1 + x
1 + 2x + 2x^2
1 + 3x + 4x^2 + 3x^3
1 + 4x + 6x^2 + 7x^3 + 5x^4
		

Crossrefs

Cf. A208515.

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*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[%]    (* A208514 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208515 *)

Formula

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

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

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