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

Author

Clark Kimberling, Mar 05 2012

Keywords

• nonn
• tabl

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

Examples

			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)
		

Crossrefs

Cf. A209135, A208510.

Programs

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 *)

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)