A209136 Triangle of coefficients of polynomials v(n,x) jointly generated with A209135; see the Formula section.

1, 1, 3, 1, 8, 5, 1, 15, 23, 11, 1, 24, 66, 66, 21, 1, 35, 150, 240, 165, 43, 1, 48, 295, 678, 747, 404, 85, 1, 63, 525, 1631, 2547, 2157, 947, 171, 1, 80, 868, 3500, 7246, 8560, 5864, 2182, 341, 1, 99, 1356, 6888, 18126, 28018, 26592, 15318, 4929, 683

Offset

1,3

Comments

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 given by (1, 0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 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..55.

Formula

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

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 3, 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;
  1,  3;
  1,  8,  5;
  1, 15, 23, 11;
  1, 24, 66, 66, 21;
First three polynomials v(n,x):
  1
  1 + 3x
  1 + 8x + 5x^2.
From Philippe Deléham, Apr 11 2012: (Start)
(1, 0, 2/3, 1/3, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   3,   0;
  1,   8,   5,   0;
  1,  15,  23,  11,   0;
  1,  24,  66,  66,  21,   0;
  1,  35, 150, 240, 165,  43,   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_] := 2 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[%]    (* A209135 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209136 *)

Crossrefs

Cf. A209135, A208510.

Sequence in context: A068437 A016477 A172157 * A206800 A258205 A258018

Adjacent sequences: A209133 A209134 A209135 * A209137 A209138 A209139

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved