A209130 Triangle of coefficients of polynomials v(n,x) jointly generated with A102756; see the Formula section.

1, 1, 2, 1, 5, 3, 1, 9, 12, 5, 1, 14, 31, 27, 8, 1, 20, 65, 89, 55, 13, 1, 27, 120, 230, 222, 108, 21, 1, 35, 203, 511, 684, 514, 205, 34, 1, 44, 322, 1022, 1777, 1834, 1125, 381, 55, 1, 54, 486, 1890, 4095, 5442, 4563, 2367, 696, 89, 1, 65, 705, 3288, 8625

Offset: 1

Author

Clark Kimberling, Mar 05 2012

Keywords

• nonn
• tabl

Comments

Top edge: (1,2,3,5,8,...) = A000045(n+1), Fibonacci numbers.

Alternating row sums: 1,-1,-1,-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 T(n,k) given by (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012

Examples

			First five rows:
  1;
  1,  2;
  1,  5,  3;
  1,  9, 12,  5;
  1, 14, 31, 27,  8;
First three polynomials v(n,x):
  1
  1 + 2x
  1 + 5x + 3x^2.
From _Philippe Deléham_, Mar 08 2012: (Start)
(1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  5,  3,  0;
  1,  9, 12,  5,  0;
  1, 14, 31, 27,  8,  0;
  1, 20, 65, 89, 55, 13, 0; ...
with row sums 1, 1, 3, 9, 27, 81, 243, 729, ... (powers of 3). (End)
		

Crossrefs

Cf. A102756, 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*u[n - 1, x] + (x + 1)*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[%]    (* A102756 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209130 *)

Formula

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

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

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

From Philippe Deléham, Mar 08 2012: (Start)

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

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

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

Sum_{k=0..n, n>=1} T(n,k)*x^k = A153881(n), A000012(n), A000244(n-1), A126473(n-1) for x = -1, 0, 1, 2 respectively. (End)