A210868 Triangle of coefficients of polynomials u(n,x) jointly generated with A210869; see the Formula section.

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 5, 3, 5, 1, 3, 5, 10, 5, 8, 1, 3, 9, 10, 20, 8, 13, 1, 4, 9, 22, 20, 38, 13, 21, 1, 4, 14, 22, 51, 38, 71, 21, 34, 1, 5, 14, 40, 51, 111, 71, 130, 34, 55, 1, 5, 20, 40, 105, 111, 233, 130, 235, 55, 89, 1, 6, 20, 65, 105, 256, 233, 474

Offset: 1

Author

Clark Kimberling, Mar 29 2012

Keywords

• nonn
• tabl

Comments

In row n the first two terms are 1 and floor(n/2), and the last two, for n>1, are F(n-1) and F(n), where F = A000045 (Fibonacci numbers).

Row sums: 1,2,4,8,16,32,...; A000079.

Alternating row sums: A151575

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

Subtriangle of the triangle given by (1, 0, -1, 0, 0, 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 02 2012

Examples

			First six rows:
  1
  1...1
  1...1...2
  1...2...2...3
  1...2...5...3....5
  1...3...5...10...5...8
First three polynomials u(n,x): 1, 1 + x, 1 + x + 2x^2.
(1, 0, -1, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
1
1, 0
1, 1, 0
1, 1, 2, 0
1, 2, 2, 3, 0
1, 2, 5, 3, 5, 0
1, 3, 5, 10, 5, 8, 0. - _Philippe Deléham_, Apr 02 2012
		

Crossrefs

Cf. A210867, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - 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[%]   (* A210866 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210867 *)

Formula

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

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

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

From Philippe Deléham, Apr 02 2012: (Start)

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

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

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