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A209142 Triangle of coefficients of polynomials v(n,x) jointly generated with A209141; see the Formula section. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

Table of n, a(n) for n=1..52.

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

EXAMPLE

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.

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. A209141, A208510.

Sequence in context: A157470 A120280 A212362 * A115754 A210600 A220308

Adjacent sequences:  A209139 A209140 A209141 * A209143 A209144 A209145

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 06 2012

STATUS

approved

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Last modified October 18 00:20 EDT 2019. Contains 328135 sequences. (Running on oeis4.)