A209142 Triangle of coefficients of polynomials v(n,x) jointly generated with A209141; see the Formula section.

1, 2, 2, 4, 7, 3, 8, 20, 17, 5, 16, 52, 65, 37, 8, 32, 128, 210, 176, 75, 13, 64, 304, 616, 679, 428, 146, 21, 128, 704, 1696, 2312, 1921, 971, 276, 34, 256, 1600, 4464, 7240, 7449, 4970, 2097, 511, 55, 512, 3584, 11360, 21344, 26146, 21622, 12056

Offset: 1

Author

Clark Kimberling, Mar 06 2012

Keywords

• nonn
• tabl

Comments

Each row begins with a power of 2 and ends with a Fibonacci number.

Alternating row sums: 1,0,0,0,0,0,0,0,0,0,0,...

For a discussion and guide to related arrays, see A208510.

As triangle T(n,k) with 0<=k<=n, it is (2, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012

Examples

			First five rows:
1
2....2
4....7....3
8....20...17...5
16...52...65...37...8
First three polynomials v(n,x): 1, 2 + 2x, 4 + 7x + 3x^2.
		

Crossrefs

Cf. A209141, A208510.

Programs

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 + 1)*u[n - 1, x] + (x + 1)*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[%]    (* A209141 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209142 *)

Formula

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

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

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

As triangle T(n,k), 0<=k<=n : T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 07 2012

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