A208756 Triangle of coefficients of polynomials v(n,x) jointly generated with A208755; see the Formula section.

1, 0, 2, 0, 1, 4, 0, 1, 3, 8, 0, 1, 3, 9, 16, 0, 1, 3, 11, 23, 32, 0, 1, 3, 13, 31, 57, 64, 0, 1, 3, 15, 39, 87, 135, 128, 0, 1, 3, 17, 47, 121, 227, 313, 256, 0, 1, 3, 19, 55, 159, 339, 579, 711, 512, 0, 1, 3, 21, 63, 201, 471, 933, 1431, 1593, 1024, 0, 1, 3, 23, 71

Offset

1,3

Comments

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

As triangle T(n,k) with 0<=k<=n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 02 2012

Links

Table of n, a(n) for n=1..71.

Formula

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

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

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

As triangle with 0<=k<=n : G.f.: (1-x+y*x)/(1-(1+y)*x-(2*y^2-y)*x^2). - Philippe Deléham, Mar 02 2012

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 02 2012

Example

First five rows:
1
0...2
0...1...4
0...1...3...8
0...1...3...9...16
First five polynomials v(n,x):
1
2x
x + 4x^2
x + 3x^2 + 8x^3
x + 3x^2 + 9x^3 + 16^4

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := 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[%]    (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208756 *)

Crossrefs

Cf. A208755, A208510.

Sequence in context: A372873 A212206 A247489 * A259873 A121462 A349706

Adjacent sequences: A208753 A208754 A208755 * A208757 A208758 A208759

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 01 2012

Status

approved