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

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A055870 Signed Fibonomial triangle. 26
1, 1, -1, 1, -1, -1, 1, -2, -2, 1, 1, -3, -6, 3, 1, 1, -5, -15, 15, 5, -1, 1, -8, -40, 60, 40, -8, -1, 1, -13, -104, 260, 260, -104, -13, 1, 1, -21, -273, 1092, 1820, -1092, -273, 21, 1, 1, -34, -714, 4641, 12376, -12376, -4641, 714, 34, -1, 1, -55, -1870, 19635, 85085, -136136, -85085, 19635, 1870, -55, -1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row n+1 (n >= 1) of the signed triangle lists the coefficients of the recursion relation for the n-th power of Fibonacci numbers A000045: sum(a(n+1,m)*(F(k-m))^n,m=0..n+1) = 0, k >= n+1; inputs: (F(k))^n, k=0..n.

The inverse of the row polynomial p(n,x) := sum(a(n,m)*x^m,m=0..n) is the g.f. for the column m=n-1 of the Fibonomial triangle A010048.

The row polynomials p(n,x) factorize according to p(n,x)=G(n-1)*p(n-2,-x), with inputs p(0,x)= 1, p(1,x)= 1-x and G(n) := 1-L(n)*x+(-1)^n*x^2, with L(n)=A000032(n) (Lucas). (Derived from Riordan's result and Knuth's exercise).

The row polynomials are the characteristic polynomials of product of the binomial matrix binomial(i,j) and the exchange matrix J_n (matrix with 1's on the antidiagonal, 0 elsewhere). - Paul Barry, Oct 05 2004

REFERENCES

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, pp. 84-5 and 492.

LINKS

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

A. T. Benjamin, S. S. Plott, A combinatorial approach to fibonomial coefficients, Fib. Quart. 46/47 (1) (2008/9) 7-9.

A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.

H. W. Gould, Extensions of the Hermite g.c.d. theorems for binomial coefficients, Fib Quart. 33 (1995) 386.

Ron Knott The Fibonomials

E. Kilic, The generalized Fibonomial matrix, Eur. J. Combinat. 31 (1) (2010) 193-209.

A. K. Kwasniewski, Fibonomial cumulative connection constants, arXiv:math/0406006 [math.CO], 2004-2009.

Ewa Krot, An introduction to finite fibonomial calculus, Centr. Eur. J. Math. 2 (5) (2004) 754.

J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.

J. Seibert, P. Trojovsky, On some identities for the Fibonomial coefficients, Math. Slov. 55 (2005) 9-19.

P. Trojovsky, On some identities for the Fibonomial coefficients..., Discr. Appl. Math. 155 (15) (2007) 2017

FORMULA

a(n, m)=(-1)^floor((m+1)/2)*A010048(n, m). A010048(n, m)=: fibonomial(n, m).

G.f. for column m: (-1)^floor((m+1)/2)*x^m/p(m+1, x) with the row polynomial of the (signed) triangle: p(n, x) := sum(a(n, m)*x^m, m=0..n).

EXAMPLE

Row polynomial for n=4: p(4,x)=1-3*x-6*x^2+3*x^3+x^4= (1+x-x^2)*(1-4*x-x^2). 1/p(4,x) is G.f. for A010048(n+3,3), n >= 0: {1,3,15,60,...}= A001655(n).

n=3: 1*(F(k))^3 - 3*(F(k-1))^3 - 6*(F(k-2))^3 + 3*(F(k-3))^3 + 1*(F(k-4))^3 = 0, k >= 4; inputs: (F(k))^3, k=0..3.

The triangle begins:

n\m 0   1     2    3     4      5     6    7   8   9

0   1

1   1  -1

2   1  -1    -1

3   1  -2    -2    1

4   1  -3    -6    3     1

5   1  -5   -15   15     5     -1

6   1  -8   -40   60    40     -8    -1

7   1 -13  -104  260   260   -104   -13    1

8   1 -21  -273 1092  1820  -1092  -273   21   1

9   1 -34  -714 4641 12376 -12376 -4641  714  34  -1

... [Wolfdieter Lang, Aug 06 2012; a(7,1) corrected, Oct 10 2012]

MAPLE

A055870 := proc(n, k)

    (-1)^floor((k+1)/2)*A010048(n, k) ;

end proc: # R. J. Mathar, Jun 14 2015

MATHEMATICA

a[n_, m_] := {1, -1, -1, 1}[[Mod[m, 4] + 1]] * Product[ Fibonacci[n - j + 1] / Fibonacci[j], {j, 1, m}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 05 2013 *)

CROSSREFS

Cf. A010048, A000032, A000045, A001654-8, A056565-7. Row sums (signed): A055871, (unsigned) A056569.

Cf. A051159.

Central column: A003268.

Sequence in context: A155865 A156133 A010048 * A088459 A007799 A122888

Adjacent sequences:  A055867 A055868 A055869 * A055871 A055872 A055873

KEYWORD

easy,sign,tabl

AUTHOR

Wolfdieter Lang, Jul 10 2000

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy .

Last modified August 17 07:13 EDT 2017. Contains 290635 sequences.