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

1,3

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

Links

Table of n, a(n) for n=1..70.

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)

Example

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.

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 *)

Crossrefs

Cf. A208328.

Sequence in context: A349728 A261163 A292244 * A283025 A089598 A117139

Adjacent sequences: A208326 A208327 A208328 * A208330 A208331 A208332

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved