A208748 Triangle of coefficients of polynomials v(n,x) jointly generated with A208747; see the Formula section.

1, 0, 4, 0, 2, 12, 0, 2, 8, 40, 0, 2, 8, 40, 128, 0, 2, 8, 48, 160, 416, 0, 2, 8, 56, 208, 640, 1344, 0, 2, 8, 64, 256, 928, 2432, 4352, 0, 2, 8, 72, 304, 1248, 3840, 9088, 14080, 0, 2, 8, 80, 352, 1600, 5504, 15616, 33280, 45568, 0, 2, 8, 88, 400, 1984, 7424

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, ...) DELTA (4, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

Links

Table of n, a(n) for n=1..62.

Formula

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

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

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

T(n,k) = 2^k*A208343(n,k). - Philippe Deléham, Mar 05 2012

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) = 1, T(2,0) = T(3,0) = 0, T(2,1) = 4, T(3,1) = 2, T(3,2) = 12, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 14 2012

G.f.: (-1+x-2*x*y)*x*y/(-1+x+2*x*y-2*x^2*y+4*x^2*y^2). - R. J. Mathar, Aug 11 2015

Example

First five rows:
1
0...4
0...2...12
0...2...8...40
0...2...8...40...128
First five polynomials v(n,x):
1
4x
2x + 12x^2
2x + 8x^2 + 40x^3
2x + 8x^2 + 40x^3 + 128x^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] + 2 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[%]    (* A208747 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208748 *)

Crossrefs

Cf. A208747, A208510.

Sequence in context: A282279 A208333 A279413 * A134895 A318468 A346492

Adjacent sequences: A208745 A208746 A208747 * A208749 A208750 A208751

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 02 2012

Status

approved