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

Author

Clark Kimberling, Mar 02 2012

Keywords

• nonn
• tabl

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

Examples

			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
		

Crossrefs

Cf. A208763, A208510.

Programs

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 *)

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