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A210747 Triangle of coefficients of polynomials u(n,x) jointly generated with A210748; see the Formula section. 3
1, 2, 3, 4, 9, 8, 7, 24, 33, 21, 12, 54, 109, 111, 55, 20, 114, 297, 435, 355, 144, 33, 228, 736, 1383, 1606, 1098, 377, 54, 441, 1697, 3912, 5813, 5625, 3316, 987, 88, 831, 3723, 10158, 18419, 22779, 18962, 9837, 2584, 143, 1536, 7859, 24798 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row n starts with -1+F(n+2) and ends with F(2n), where F=A000045 (Fibonacci numbers).

Row sums: A002450

Alternating row sums: A077925

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

LINKS

Table of n, a(n) for n=1..49.

FORMULA

u(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,

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

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

EXAMPLE

First five rows:

1

2....3

4....9....8

7....24...33....21

12...54...109...111...55

First three polynomials u(n,x): 1, 2+ 3x, 4 + 9x + 8x^2.

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%] (* A210747 *)

Table[Expand[v[n, x]], {n, 1, z}]

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

TableForm[cv]

Flatten[%] (* A210748 *)

Table[u[n, x] /. x -> 1, {n, 1, z}] (* A002450 *)

Table[v[n, x] /. x -> 1, {n, 1, z}] (* A002450 *)

Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)

Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)

CROSSREFS

Cf. A208510.

Sequence in context: A307404 A307405 A115305 * A353239 A351497 A329425

Adjacent sequences: A210744 A210745 A210746 * A210748 A210749 A210750

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 25 2012

STATUS

approved

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Last modified November 29 03:18 EST 2022. Contains 358422 sequences. (Running on oeis4.)