A208338 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section.

1, 1, 1, 1, 2, 3, 1, 3, 7, 7, 1, 4, 12, 20, 17, 1, 5, 18, 40, 57, 41, 1, 6, 25, 68, 129, 158, 99, 1, 7, 33, 105, 243, 399, 431, 239, 1, 8, 42, 152, 410, 824, 1200, 1160, 577, 1, 9, 52, 210, 642, 1506, 2692, 3528, 3089, 1393, 1, 10, 63, 280, 952, 2532, 5290

Offset: 1

Author

Clark Kimberling, Feb 27 2012

Keywords

• nonn
• tabl

Comments

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012

Examples

			First five rows:
  1;
  1,  1;
  1,  2,  3;
  1,  3,  7,  7;
  1,  4, 12, 20, 17;
First five polynomials u(n,x):
  1
  1 +  x
  1 + 2x +  3x^2
  1 + 3x +  7x^2 +  7x^3
  1 + 4x + 12x^2 + 20x^3 + 17x^4
From _Philippe Deléham_, Apr 09 2012: (Start)
(1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  3,  0;
  1,  3,  7,  7,  0;
  1,  4, 12, 20, 17,  0;
  1,  5, 18, 40, 57, 41,  0; (End)
		

Crossrefs

Cf. A208339.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*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[%]   (* A208338 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208339 *)

Formula

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

v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),

where u(1,x)=1, v(1,x)=1.

From Philippe Deléham, Apr 09 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) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)