A209696 Triangle of coefficients of polynomials v(n,x) jointly generated with A209695; see the Formula section.

1, 1, 3, 1, 6, 7, 1, 9, 23, 17, 1, 12, 48, 76, 41, 1, 15, 82, 204, 233, 99, 1, 18, 125, 428, 765, 682, 239, 1, 21, 177, 775, 1907, 2649, 1935, 577, 1, 24, 238, 1272, 4010, 7656, 8680, 5368, 1393, 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363

Offset: 1

Author

Clark Kimberling, Mar 13 2012

Keywords

• nonn
• tabl

Comments

Alternating row sums: 1,-2,2,-2,2,-2,2,-2,...

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

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012

Mirror image of triangle in A054458. - Philippe Deléham, Mar 24 2012

Examples

			First five rows:
  1;
  1,  3;
  1,  6,  7;
  1,  9, 23, 17;
  1, 12, 48, 76, 41;
First three polynomials v(n,x):
  1
  1 + 3x
  1 + 6x + 7x^2.
From _Philippe Deléham_, Mar 24 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, ...) begins:
  1;
  1,  0;
  1,  3,  0;
  1,  6,  7,  0;
  1,  9, 23, 17,  0;
  1, 12, 48, 76, 41,  0; (End)
		

Crossrefs

Cf. A054458, A209695, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 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[%]    (* A209695 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209696 *)

Formula

u(n,x) = 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, Mar 24 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

G.f.: (1-2*y*x-y^2*x^2)/(1-x-2*y*x-y*x^2-y^2*x^2).

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) + 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)