A209125 Triangle of coefficients of polynomials u(n,x) jointly generated with A164975; see the Formula section.

1, 2, 1, 3, 4, 2, 5, 9, 9, 4, 8, 20, 25, 20, 8, 13, 40, 65, 65, 44, 16, 21, 78, 150, 190, 162, 96, 32, 34, 147, 331, 490, 521, 392, 208, 64, 55, 272, 697, 1192, 1473, 1368, 928, 448, 128, 89, 495, 1425, 2745, 3888, 4185, 3480, 2160, 960, 256, 144, 890

Offset

1,2

Comments

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, -1, 0, 0, 0, 0, 0, 0, 0, ....) DELTA (0, 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..57.

Formula

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

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

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

Example

First five rows:
  1;
  2,  1;
  3,  4,  2;
  5,  9,  9,  4;
  8, 20, 25, 20,  8;
First three polynomials u(n,x):
  1
  2 + x
  3 + 4x + 2x^2
From Philippe Deléham, Mar 21 2012: (Start)
(1, 1, -1, 0, 0, ...) DELTA (0, 1, 1, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  3,  4,  2,  0;
  5,  9,  9,  4, 0;
  8, 20, 25, 20, 8, 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_] := 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[%]    (* A209125 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A164975 *)

Crossrefs

Cf. A164975, A208510.

Sequence in context: A026249 A130527 A026366 * A209137 A269752 A122164

Adjacent sequences: A209122 A209123 A209124 * A209126 A209127 A209128

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved