A209134 Triangle of coefficients of polynomials v(n,x) jointly generated with A209133; see the Formula section.

1, 0, 4, 0, 4, 10, 0, 4, 18, 28, 0, 4, 26, 72, 76, 0, 4, 34, 132, 256, 208, 0, 4, 42, 208, 572, 864, 568, 0, 4, 50, 300, 1056, 2272, 2808, 1552, 0, 4, 58, 408, 1740, 4800, 8496, 8896, 4240, 0, 4, 66, 532, 2656, 8880, 20208, 30432, 27648, 11584, 0, 4, 74

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, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 10 2012

Links

Table of n, a(n) for n=1..58.

Formula

u(n,x)=u(n-1,x)+(x+1)*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) = T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 4 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Apr 10 2012

Example

First five rows:
1
0...4
0...4...10
0...4...18...28
0...4...26...72...76
First three polynomials v(n,x): 1, 4x, 4x + 10x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*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[%]    (* A209133 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209134 *)

Crossrefs

Cf. A209133, A208510.

Sequence in context: A156732 A200341 A101980 * A281297 A058536 A371377

Adjacent sequences: A209131 A209132 A209133 * A209135 A209136 A209137

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved