A209139 Triangle of coefficients of polynomials u(n,x) jointly generated with A209140; see the Formula section.

1, 2, 1, 3, 5, 3, 5, 12, 15, 7, 8, 27, 45, 42, 17, 13, 55, 119, 151, 116, 41, 21, 108, 282, 458, 480, 315, 99, 34, 205, 630, 1228, 1631, 1467, 845, 239, 55, 381, 1343, 3054, 4849, 5502, 4358, 2244, 577, 89, 696, 2769, 7173, 13218, 17895, 17838, 12666

Offset

1,2

Comments

Column 1: A000045 (Fibonacci numbers).

Alternating row sums: 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 given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012

Links

Table of n, a(n) for n=1..53.

Formula

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

v(n,x) = (x+1)*u(n-1,x) + 2x*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.: (1-2*y*x-y*x^2-y^2*x^2)/(1-x-x^2-2*y*x-y^2*x^2).

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + 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)

Example

First five rows:
  1;
  2,  1;
  3,  5,  3;
  5, 12, 15,  7;
  8, 27, 45, 42, 17;
First three polynomials u(n,x):
  1
  2 + x
  3 + 5x + 3x^2
From Philippe Deléham, Apr 11 2012: (Start)
(1, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,   0;
   3,  5,   3,   0;
   5, 12,  15,   7,   0;
   8, 27,  45,  42,  17,  0;
  13, 55, 119, 151, 116, 41, 0; (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 + 1)*u[n - 1, x] + 2 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[%]    (* A209139 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209140 *)

Crossrefs

Cf. A209140, A208510.

Sequence in context: A258236 A258244 A258248 * A257161 A253676 A182939

Adjacent sequences: A209136 A209137 A209138 * A209140 A209141 A209142

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved