A208766 Triangle of coefficients of polynomials v(n,x) jointly generated with A208765; see the Formula section.

1, 1, 3, 1, 6, 7, 1, 9, 21, 19, 1, 12, 42, 76, 47, 1, 15, 70, 190, 235, 123, 1, 18, 105, 380, 705, 738, 311, 1, 21, 147, 665, 1645, 2583, 2177, 803, 1, 24, 196, 1064, 3290, 6888, 8708, 6424, 2047, 1, 27, 252, 1596, 5922, 15498, 26124, 28908, 18423

Offset

1,3

Comments

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

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -4/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 20 2012

Links

Table of n, a(n) for n=1..54.

Formula

u(n,x) = u(n-1,x) + 2x*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, Mar 20 2012: (Start)

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

G.f.: (1-x-y*x+3*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*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) + 4*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. (End)

Example

First five rows:
  1;
  1,  3;
  1,  6,  7;
  1,  9, 21, 19;
  1, 12, 42, 76, 47;
First five polynomials v(n,x):
  1
  1 +  3x
  1 +  6x +  7x^2
  1 +  9x + 21x^2 + 19x^3
  1 + 12x + 42x^2 + 76x^3 + 47x^4
From Philippe Deléham, Mar 20 2012: (Start)
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 3, -2/3, -4/3, 0, 0, ...) begins:
  1;
  1,  0;
  1,  3,  0;
  1,  6,  7,  0;
  1,  9, 21, 19,  0;
  1, 12, 42, 76, 47,  0; (End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*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[%]     (* A208765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A208766 *)

Crossrefs

Cf. A208765, A208510.

Sequence in context: A198614 A239385 A124929 * A259454 A209696 A338995

Adjacent sequences: A208763 A208764 A208765 * A208767 A208768 A208769

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved