login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A132148 Triangular array T(n,k) = C(n,k)*Lucas(n-k), 0 <= k <= n. 3
2, 1, 2, 3, 2, 2, 4, 9, 3, 2, 7, 16, 18, 4, 2, 11, 35, 40, 30, 5, 2, 18, 66, 105, 80, 45, 6, 2, 29, 126, 231, 245, 140, 63, 7, 2, 47, 232, 504, 616, 490, 224, 84, 8, 2, 76, 423, 1044, 1512, 1386, 882, 336, 108, 9, 2, 123, 760, 2115, 3480, 3780, 2772, 1470, 480, 135, 10, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The row polynomials L(n,x) = sum {k = 0 .. n} C(n,k)*Lucas(n-k)*x^k satisfy L(n,x)* F(n,x) = F(2n,x), where F(n,x) = sum {k = 0 .. n} C(n,k)*Fibonacci(n-k)*x^k.

Other identities and formulas include: L(n+1,x)^2 - L(n,x)*L(n+2,x) = -5*(x^2 + x - 1)^n; L(n+1,x) - (x^2 + x - 1)*L(n-1,x) = 5*F(n,x) for n >= 1; L(2n,x) - 2*(x^2 + x - 1)^n = 5*F(n,x)^2; L(n,2x) = sum { k = 0 .. n} C(n,k)*L(n-k,x)*x^k; L(n,3x) = sum { k = 0 .. n} C(n,k)*L(n-k,2x)*x^k etc;

Sum {k = 0 .. n} C(n,k)*L(k,x)*F(n-k,x) = 2^n F(n,x); Row sums: L(n,1) = Lucas(2n); Alternating row sums: L(n,-1) = (-1)^n Lucas(n); L(n,1/phi) = (-1)^n L(n,-phi) = sqrt(5)^n for n >= 1, where phi = (1+sqrt(5))/2.

The polynomials L(n,-x) satisfy a Riemann hypothesis: the zeros of L(n,-x) lie on the vertical line Re x = 1/2 in the complex plane.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

G.f.: (2 - (2x + 1)*t)/(1 - (2x + 1)*t + (x^2 + x - 1)*t^2) = 2 + (1 + 2x)*t + (3 + 2x + 2x^2)*t^2 + (4 + 9x + 3x^2 + 2x^3)*t^3 + ... .

EXAMPLE

Triangle starts

2;

1, 2;

3, 2, 2;

4, 9, 3, 2;

MAPLE

with(combinat): lucas := n -> fibonacci(n-1) + fibonacci(n+1): T := (n, k) -> binomial(n, k)*lucas(n-k): for n from 0 to 10 do seq( T(n, k), k = 0..n) od;

MATHEMATICA

Flatten[Table[Binomial[n, k]LucasL[n-k], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Nov 06 2011 *)

CROSSREFS

Cf. A000032, A000045, A094440.

Sequence in context: A111725 A112218 A172366 * A237829 A159974 A143866

Adjacent sequences:  A132145 A132146 A132147 * A132149 A132150 A132151

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Bala, Aug 17 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 23 11:48 EST 2014. Contains 249842 sequences.