A208764 Triangle of coefficients of polynomials v(n,x) jointly generated with A208763; see the Formula section.

1, 0, 3, 0, 2, 7, 0, 2, 6, 19, 0, 2, 6, 26, 47, 0, 2, 6, 34, 78, 123, 0, 2, 6, 42, 110, 258, 311, 0, 2, 6, 50, 142, 426, 758, 803, 0, 2, 6, 58, 174, 626, 1366, 2282, 2047, 0, 2, 6, 66, 206, 858, 2134, 4594, 6558, 5259, 0, 2, 6, 74, 238, 1122, 3062, 7866, 14334

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, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -4/3, 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..64.

Formula

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

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

G.f.: (1-x+2y*x)/(1-(1+y)*x -(4*y^2-y)*x^2). - Philippe Deléham, Mar 02 2012

As triangle T(n,k), 0<=k<=n : T(n,k) = T(n-1,k) + T(n-1,k-1) + 4*T(n-2,k-2) - T(n-2,k-1) 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. - Philippe Deléham, Mar 02 2012

Example

First five rows:
1
0...3
0...2...7
0...2...6...19
0...2...6...26...47
First five polynomials v(n,x):
1
3x
2x + 7x^2
2x + 6x^2 + 19x^3
2x + 6x^2 + 26x^3 + 47x^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_] := 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[%]    (* A208763 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208764 *)

Crossrefs

Cf. A208763, A208510.

Sequence in context: A298058 A298707 A274417 * A209129 A368479 A282694

Adjacent sequences: A208761 A208762 A208763 * A208765 A208766 A208767

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved