A209138 Triangle of coefficients of polynomials v(n,x) jointly generated with A209137; see the Formula section.

1, 1, 2, 2, 4, 3, 3, 9, 10, 5, 5, 18, 28, 22, 8, 8, 35, 68, 74, 45, 13, 13, 66, 154, 210, 177, 88, 21, 21, 122, 331, 541, 574, 397, 167, 34, 34, 222, 686, 1302, 1656, 1446, 850, 310, 55, 55, 399, 1382, 2982, 4404, 4614, 3434, 1758, 566, 89, 89, 710, 2723

Offset: 1

Author

Clark Kimberling, Mar 05 2012

Keywords

• nonn
• tabl

Comments

Every row begins and ends with a Fibonacci number (A000045).

u(n,1) = n-th row sum = 3^(n-1).

Alternating row sums: 1,-1,1,-1,1,-1,1,-1,1,-1,...

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
  1;
  1,  2;
  2,  4,  3;
  3,  9, 10,  5;
  5, 18, 28, 22,  8;
First three polynomials v(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.
From _Philippe Deléham_, Apr 11 2012: (Start)
Triangle in A185081 begins:
  1;
  0,  1;
  0,  1,  2;
  0,  2,  4,  3;
  0,  3,  9, 10,  5;
  0,  5, 18, 28, 22,  8;
  ... (End)
		

Crossrefs

Cf. A209137, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A209137 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209138 *)

Formula

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

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

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

From Philippe Deléham, Apr 11 2012: (Start)

T(n,k) = A185081(n,k+1).

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