A209128 Triangle of coefficients of polynomials u(n,x) jointly generated with A209129; see the Formula section.

1, 2, 1, 2, 4, 3, 2, 6, 12, 7, 2, 8, 22, 32, 17, 2, 10, 34, 70, 86, 41, 2, 12, 48, 124, 216, 228, 99, 2, 14, 64, 196, 428, 644, 600, 239, 2, 16, 82, 288, 744, 1408, 1876, 1568, 577, 2, 18, 102, 402, 1188, 2664, 4476, 5364, 4074, 1393, 2, 20, 124, 540, 1786

Offset

1,2

Comments

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, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

Row sums are powers of 3 (A000244). - Philippe Deléham, Mar 21 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) + 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+x^2-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).

T(n,k) = T(n-1,k-1) + 2*T(n-1,k-1) - T(n-2,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,  4,  3;
  2,  6, 12,  7;
  2,  8, 22, 32, 17;
First three polynomials u(n,x):
  1
  2 + x
  2 + 4x + 3x^2
From Philippe Deléham, Mar 21 2012: (Start)
(1, 1, -2, 1, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  2,  4,  3,  0;
  2,  6, 12,  7,  0;
  2,  8, 22, 32, 17,  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] + 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[%]    (* A209128 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209129 *)

Crossrefs

Cf. A209129, A208510.

Sequence in context: A099312 A117505 A331499 * A209131 A165053 A302982

Adjacent sequences: A209125 A209126 A209127 * A209129 A209130 A209131

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved