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A210872 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section. 3
1, 0, 1, 0, 2, 1, 0, 1, 5, 1, 0, 1, 4, 9, 1, 0, 1, 3, 12, 14, 1, 0, 1, 3, 9, 29, 20, 1, 0, 1, 3, 8, 27, 60, 27, 1, 0, 1, 3, 8, 22, 74, 111, 35, 1, 0, 1, 3, 8, 21, 63, 181, 189, 44, 1, 0, 1, 3, 8, 21, 56, 178, 399, 302, 54, 1, 0, 1, 3, 8, 21, 55, 154, 474, 806, 459, 65, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Column 1: 1,0,0,0,0,0,0,0,0,...

Row sums: A000225 (-1+2^n)

Alternating row sums:  (-1)*A077973

Limiting row:  0,1,3,8,21,..., even-indexed Fibonacci numbers

If the term in row n and column k is written as U(n,k), then U(n,n-1)=A000096 and U(n,n-2)=A086274.

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

LINKS

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

FORMULA

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

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

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

Also, u(n,x)=2x*u(n-1,x)+(x-x^2)*u(n-2,x)+x, where u(2,x)=x.

EXAMPLE

First six rows:

1

0...1

0...2...1

0...1...5...1

0...1...4...9....1

0...1...3...12...14...1

First three polynomials u(n,x): 1, x, 2x + x^2.

MATHEMATICA

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

u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;

v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;

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[%]    (* A210872 *)

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

TableForm[cv]

Flatten[%]    (* A210873 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)

Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)

Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)

CROSSREFS

Cf. A210873, A208510.

Sequence in context: A075374 A293024 A292948 * A292973 A220235 A066603

Adjacent sequences:  A210869 A210870 A210871 * A210873 A210874 A210875

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 29 2012

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)