A209129 Triangle of coefficients of polynomials v(n,x) jointly generated with A209128; see the Formula section.

1, 0, 3, 0, 2, 7, 0, 2, 8, 17, 0, 2, 10, 28, 41, 0, 2, 12, 42, 88, 99, 0, 2, 14, 58, 154, 262, 239, 0, 2, 16, 76, 240, 524, 752, 577, 0, 2, 18, 96, 348, 908, 1692, 2104, 1393, 0, 2, 20, 118, 480, 1440, 3224, 5260, 5776, 3363, 0, 2, 22, 142, 638, 2148, 5544

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, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

Links

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

Formula

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

v(n,x)=x*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) - T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 21 2012

G.f.: (-1+x-x*y)*x*y/(-1+x+2*x*y-x^2*y+x^2*y^2). - R. J. Mathar, Aug 12 2015

Example

First five rows:
1
0...3
0...2....7
0...2....8...17
0...2...10...28...41
First three polynomials v(n,x): 1, 3x, 2x + 7x^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_] := 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[%]    (* A209128 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209129 *)

Crossrefs

Cf. A209128, A208510.

Sequence in context: A298707 A274417 A208764 * A368479 A282694 A011075

Adjacent sequences: A209126 A209127 A209128 * A209130 A209131 A209132

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 05 2012

Status

approved