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

Author

Wolfdieter Lang, Jul 10 2000

Keywords

• easy
• sign
• tabl

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_{m=0..n+1} T(n+1,m)*(Fibonacci(k-m))^n = 0, k >= n+1; inputs: (Fibonacci(k))^n, k=0..n.

The inverse of the row polynomial p(n,x) := Sum_{m=0..n} T(n,m)*x^m 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 - A000032(n)*x + (-1)^n*x^2. (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

Examples

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

References

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

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• 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 and 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.

Crossrefs

Cf. A000032, A000045, A001654, A001655, A001656, A001657, A001658, A010048, A051159, A056565, A056566, A056567.

Sums include: A055871 (signed row), A056569 (row).

Central column: A003268.

Cf. A383715.

Programs

Magma

Fibonomial:= func< n,k | k eq 0 select 1 else (&*[Fibonacci(n-j+1)/Fibonacci(j): j in [1..k]]) >;
[(-1)^Floor((k+1)/2)*Fibonomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 20 2024

Maple

A055870 := proc(n,k)
    (-1)^floor((k+1)/2)*A010048(n,k) ;
end proc: # R. J. Mathar, Jun 14 2015

Mathematica

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

SageMath

def fibonomial(n,k): return 1 if k==0 else product(fibonacci(n-j+1)/fibonacci(j) for j in range(1,k+1))
flatten([[(-1)^((k+1)//2)*fibonomial(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 20 2024

Formula

T(n, m) = (-1)^floor((m+1)/2)*A010048(n, m), where 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_{m=0..n} T(n, m)*x^m.

Sum_{k=0..n} T(n,k) * x^k = exp( -Sum_{k>=1} Fibonacci(n*k)/Fibonacci(k) * x^k/k ). - Seiichi Manyama, May 07 2025