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A162515 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = (U^n-L^n)/d, where U=(x+d)/2, L=(x-d)/2, d=(4 + x^2)^(1/2). 4
0, 1, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 1, 0, 4, 0, 3, 0, 1, 0, 5, 0, 6, 0, 1, 1, 0, 6, 0, 10, 0, 4, 0, 1, 0, 7, 0, 15, 0, 10, 0, 1, 1, 0, 8, 0, 21, 0, 20, 0, 5, 0, 1, 0, 9, 0, 28, 0, 35, 0, 15, 0, 1, 1, 0, 10, 0, 36, 0, 56, 0, 35, 0, 6, 0, 1, 0, 11, 0, 45, 0, 84, 0, 70, 0, 21, 0, 1, 1, 0, 12, 0, 55 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Row sums 0,1,1,2,3,5,... are the Fibonacci numbers, A000045.

Note that the coefficients are given in decreasing order.  - M. F. Hasler, Dec 07 2011

Essentially a mirror image of A168561. - Philippe Deléham, Dec 08 2013

LINKS

Table of n, a(n) for n=0..96.

FORMULA

P(n,x)=x*P(n-1,x)+P(n-2,x), where P(0,x)=0 and P(1,x)=1.

T(n,k)=T(n-1,k)+T(n-2,k-2) for n>=2. - Philippe Deléham, Dec 08 2013

EXAMPLE

First rows:

0 (row 0)

1

1...0

1...0...1

1...0...2...0

1...0...3...0...1

1...0...4...0...3...0

Row 6 matches P(6,x)=x^5 + 4*x^3 + 3*x.

MATHEMATICA

d = (4 + x^2)^(1/2); u = (x + d)/2; l = (x - d)/2;

f[n_] := (u^n - l^n)/d;

Factor[f[1]]  (* P(1, x) *)

Factor[f[5]]  (* P(5, x) *)

Factor[f[8]]

Factor[f[21]]

Factor[f[144]]

PROG

(PARI) P(n) =  my( d=(4 + x^2)^(1/2), U=(x+d)/2, L=(x-d)/2); Pol((U^n-L^n)/d)  \\ - M. F. Hasler, Dec 07 2011

CROSSREFS

Cf. A000045, A162514, A162516, A162517, A053119, A049310.

Sequence in context: A083280 A060689 A053119 * A175267 A108045 A143728

Adjacent sequences:  A162512 A162513 A162514 * A162516 A162517 A162518

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Jul 05 2009

STATUS

approved

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Last modified November 28 01:41 EST 2014. Contains 250286 sequences.