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

1, 2, 1, 3, 4, 2, 5, 10, 9, 3, 8, 22, 28, 18, 5, 13, 45, 74, 68, 35, 8, 21, 88, 177, 210, 154, 66, 13, 34, 167, 397, 574, 541, 331, 122, 21, 55, 310, 850, 1446, 1656, 1302, 686, 222, 34, 89, 566, 1758, 3434, 4614, 4404, 2982, 1382, 399, 55, 144, 1020

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: 1,1,1,1,1,1,1,1,1,1,1,1,...

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

Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012

Mirror image of triangle in A209138. - Philippe Deléham, Apr 11 2012

Examples

			First five rows:
  1;
  2,  1;
  3,  4,  2;
  5, 10,  9,  3;
  8, 22, 28, 18, 5;
First three polynomials u(n,x):
  1
  2 +  x
  3 + 4x + 2x^2
From _Philippe Deléham_, Apr 11 2012: (Start)
(1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  4,  2,  0;
   5, 10,  9,  3,  0;
   8, 22, 28, 18,  5,  0;
  13, 45, 74, 68, 35,  8,  0; (End)
		

Crossrefs

Cf. A209138, 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)

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

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

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