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

1, 2, 1, 4, 5, 2, 8, 16, 12, 3, 16, 44, 49, 25, 5, 32, 112, 166, 127, 50, 8, 64, 272, 504, 513, 301, 96, 13, 128, 640, 1424, 1808, 1408, 670, 180, 21, 256, 1472, 3824, 5816, 5641, 3562, 1427, 331, 34, 512, 3328, 9888, 17520, 20330, 15981, 8494, 2939

Offset

1,2

Comments

Each row begins with a power of 2 and ends with a Fibonacci number. Alternating row sums: all 1's. For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle T(n,k) given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012

Links

Table of n, a(n) for n=1..53.

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.

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) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Mar 07 2012

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

Example

First five rows:
1
2....1
4....5....2
8....16...12...3
16...44...49...25...5
First three polynomials u(n,x):  1, 2 + x, 4 + 5x + 2x^2
Triangle (1, 1, 0, 0, 0...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
4, 5, 2, 0
8, 16, 12, 3, 0
16, 44, 49, 25, 5, 0
32, 112, 166, 127, 50, 8, 0

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 *)

Crossrefs

Cf. A209142, A208510.

Sequence in context: A209154 A144332 A209153 * A038719 A376286 A125751

Adjacent sequences: A209138 A209139 A209140 * A209142 A209143 A209144

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 06 2012

Status

approved