A209126 Triangle of coefficients of polynomials u(n,x) jointly generated with A209127; see the Formula section.

1, 2, 1, 2, 3, 2, 2, 5, 7, 3, 2, 7, 14, 13, 5, 2, 9, 23, 32, 25, 8, 2, 11, 34, 62, 71, 46, 13, 2, 13, 47, 105, 156, 149, 84, 21, 2, 15, 62, 163, 295, 367, 304, 151, 34, 2, 17, 79, 238, 505, 767, 827, 604, 269, 55, 2, 19, 98, 332, 805, 1435, 1889, 1798, 1177, 475

Offset

1,2

Comments

u(n,n) = A000045(n), Fibonacci numbers.

Alternating row sums: 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, -2, 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, Mar 21 2012

Links

Table of n, a(n) for n=1..65.

Formula

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

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

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

From Philippe Deléham, Mar 21 2012: (Start)

As DELTA-triangle with 0 <= k <= n:

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

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

Example

First five rows:
  1;
  2,  1;
  2,  3,  2;
  2,  5,  7,  3;
  2,  7, 14, 13,  5;
First three polynomials u(n,x):
  1
  2 + x
  2 + 3x + 2x^2
From Philippe Deléham, Mar 21 2012: (Start)
(1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  2,  3,  2,  0;
  2,  5,  7,  3,  0;
  2,  7, 14, 13,  5,  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*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[%]    (* A209126 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209127 *)

Crossrefs

Cf. A209127, A208510.

Sequence in context: A155002 A182413 A296664 * A272377 A103342 A275438

Adjacent sequences: A209123 A209124 A209125 * A209127 A209128 A209129

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved