A055870 Signed Fibonomial triangle.

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

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.

Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).

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.

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

Ron Knott, The Fibonomials

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

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

Phakhinkon Phunphayap, Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes, Ph. D. Thesis, Silpakorn University (Thailand 2021).

Phakhinkon Phunphayap, Prapanpong Pongsriiam, Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients, J. Int. Seq. 21 (2018), #18.3.1.

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-François 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 * A360208 A360571 A088459

Adjacent sequences: A055867 A055868 A055869 * A055871 A055872 A055873

Keyword

easy,sign,tabl

Author

Wolfdieter Lang, Jul 10 2000

Status

approved