A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.

1, 0, 2, 0, 1, 3, 0, 1, 2, 5, 0, 1, 2, 5, 8, 0, 1, 2, 6, 10, 13, 0, 1, 2, 7, 13, 20, 21, 0, 1, 2, 8, 16, 29, 38, 34, 0, 1, 2, 9, 19, 39, 60, 71, 55, 0, 1, 2, 10, 22, 50, 86, 122, 130, 89, 0, 1, 2, 11, 25, 62, 116, 187, 241, 235, 144, 0, 1, 2, 12, 28, 75, 150, 267, 392, 468

Offset: 1

Author

Clark Kimberling, Feb 25 2012

Keywords

• nonn
• tabl

Comments

u(n,n) = A000045(n+1) (Fibonacci numbers).

n-th row sum: 2^(n-1)

As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

Examples

			First five rows:
  1;
  0, 2;
  0, 1, 3;
  0, 1, 2, 5;
  0, 1, 2, 5, 8;
First five polynomials v(n,x):
  1
     2x
      x + 3x^2
      x + 2x^2 + 5x^3
      x + 2x^2 + 5x^3 + 8x^4.
		

Crossrefs

Cf. A208343.

Cf. A084938, A000045, A000079.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
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];
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[%]  (* A208342 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208343 *)

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),

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

From Philippe Deléham, Feb 26 2012: (Start)

As triangle T(n,k) with 0 <= k <= n:

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k > n or if k < 0.

G.f.: (1-(1-y)*x)/(1-(1+y)*x+y*(1-y)*x^2).

Sum_{k=0..n} T(n,k)*x^k = (-1)*A091003(n+1), A152166(n), A000007(n), A000079(n), A055099(n), A152224(n) for x = -2, -1, 0, 1, 2, 3 respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A087205(n), A140165(n+1), A016116(n+1), A000045(n+2), A000079(n), A122367(n), A006012(n), A052961(n), A154626(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively. (End)

T(n,k) = A208748(n,k)/2^k. - Philippe Deléham, Mar 05 2012