A208329 Triangle of coefficients of polynomials v(n,x) jointly generated with A208328; see the Formula section.

1, 0, 3, 0, 2, 5, 0, 2, 4, 11, 0, 2, 4, 14, 21, 0, 2, 4, 18, 32, 43, 0, 2, 4, 22, 44, 82, 85, 0, 2, 4, 26, 56, 130, 188, 171, 0, 2, 4, 30, 68, 186, 324, 438, 341, 0, 2, 4, 34, 80, 250, 492, 834, 984, 683, 0, 2, 4, 38, 92, 322, 692, 1374, 2028, 2202, 1365, 0, 2, 4, 42

Offset: 1

Author

Clark Kimberling, Feb 26 2012

Keywords

• nonn
• tabl

Comments

Row sums, u(n,1): A000129

Row sums, v(n,1): A001333

As triangle T(n,k) with 0 <= k <= n, it is (0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 27 2012

Examples

			First five rows:
  1;
  0,  3;
  0,  2,  5;
  0,  2,  4, 11;
  0,  2,  4, 14, 21;
First five polynomials u(n,x):
  1
     3x
     2x + 5x^2
     2x + 4x^2 + 11x^3
     2x + 4x^2 + 14x^3 + 21x^4.
		

Crossrefs

Cf. A208328.

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_] := 2 x*u[n - 1, x] + 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[%]  (* A208328 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208329 *)

Formula

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

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

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

From Philippe Deléham, Feb 27 2012: (Start)

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

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3 and T(n,k) = 0 if k < 0 or if k > n.

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

Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A108411(n+1), A000007(n), A001333(n+1) for x = -1, 0, 1 respectively. (End)