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

1,3

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

Links

Table of n, a(n) for n=1..60.

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)

Example

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)

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 *)

Crossrefs

Cf. A102756, A208510.

Sequence in context: A104731 A240192 A264751 * A330381 A210792 A105728

Adjacent sequences: A209127 A209128 A209129 * A209131 A209132 A209133

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved