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A210869 Triangle of coefficients of polynomials v(n,x) jointly generated with A210868; see the Formula section. 3
1, 0, 2, 1, 0, 3, 0, 3, 0, 5, 1, 0, 7, 0, 8, 0, 4, 0, 15, 0, 13, 1, 0, 12, 0, 30, 0, 21, 0, 5, 0, 31, 0, 58, 0, 34, 1, 0, 18, 0, 73, 0, 109, 0, 55, 0, 6, 0, 54, 0, 162, 0, 201, 0, 89, 1, 0, 25, 0, 145, 0, 344, 0, 365, 0, 144, 0, 7, 0, 85, 0, 361, 0, 707, 0, 655, 0, 233, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row n starts with 1 or 0 and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Row sums:  1,2,4,8,16,32,... (A000079)

Alternating row sums: 1, -2, 4, -8, 16,... (A122803)

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

LINKS

Table of n, a(n) for n=1..80.

FORMULA

u(n,x)=u(n-1,x)+x*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) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Apr 02 2012

EXAMPLE

First six rows:

1

0...2

1...0...3

0...3...0...5

1...0...7...0....8

0...4...0...15...0...13

First three polynomials v(n,x): 1, 2x, 1 + 3x^2

MATHEMATICA

u[1, x_]:=1; v[1, x_]:=1; z=14;

u[n_, x_]:=u[n-1, x]+x*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[%]   (* A210868 *)

cv=Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]   (* A210869 *)

Table[u[n, x]/.x->1, {n, 1, z}]   (* A000072 *)

Table[v[n, x]/.x->1, {n, 1, z}]   (* A000072 *)

Table[u[n, x]/.x->-1, {n, 1, z}]  (* A151575 *)

Table[v[n, x]/.x->-1, {n, 1, z}]  (* A122803 *)

CROSSREFS

Cf. A210868, A208510.

Sequence in context: A128179 A178780 A058558 * A123973 A098493 A058560

Adjacent sequences:  A210866 A210867 A210868 * A210870 A210871 A210872

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 29 2012

STATUS

approved

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Last modified October 16 03:28 EDT 2019. Contains 328040 sequences. (Running on oeis4.)