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

1, 2, 1, 3, 5, 3, 4, 14, 16, 5, 5, 30, 54, 39, 11, 6, 55, 144, 171, 98, 21, 7, 91, 329, 561, 503, 229, 43, 8, 140, 672, 1534, 1928, 1380, 532, 85, 9, 204, 1260, 3690, 6106, 6084, 3636, 1203, 171, 10, 285, 2208, 8058, 16852, 21890, 18060, 9249, 2694

Offset

1,2

Comments

For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 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..54.

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+x^2-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(2,1) = 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;
  4, 14, 16,  5;
  5, 30, 54, 39, 11;
First three polynomials u(n,x):
  1
  2 + x
  3 + 5x + 3x^2
From Philippe Deléham, Apr 11 2012: (Start)
(1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, ...) begins:
  1;
  1,  0;
  2,  1,  0;
  3,  5,  3,  0;
  4, 14, 16,  5,  0;
  5, 30, 54, 39, 11,  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. A209136, A208510.

Sequence in context: A069931 A209152 A209158 * A258236 A258244 A258248

Adjacent sequences: A209132 A209133 A209134 * A209136 A209137 A209138

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved