A383715 -id:A383715 - cp's OEIS frontend

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

Wolfdieter Lang, Jul 10 2000

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

			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]

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

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.

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.

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

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

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 *)

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

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

A099927 Pellonomial triangle P(k,n) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 30, 12, 1, 1, 29, 174, 174, 29, 1, 1, 70, 1015, 2436, 1015, 70, 1, 1, 169, 5915, 34307, 34307, 5915, 169, 1, 1, 408, 34476, 482664, 1166438, 482664, 34476, 408, 1, 1, 985, 200940, 6791772, 39618670, 39618670, 6791772, 200940, 985, 1

Offset: 0

Ralf Stephan, Nov 03 2004

Also (signed) coefficients of solutions to 0 = Sum[i=0..k+1, x(i)*Pell(m+i)^k ].

Sagan and Savage give two combinatorial interpretations for entry T(n,k) in terms of statistics on integer partitions fitting inside a k x (n-k) rectangle. They also relate the values T(n,k) to q-binomial coefficients evaluated at q = -(3 + 2*sqrt(2)). - Peter Bala, Mar 15 2013

			Triangle starts:
  1;
  1,   1;
  1,   2,    1;
  1,   5,    5,     1;
  1,  12,   30,    12,     1;
  1,  29,  174,   174,    29,    1;
  1,  70, 1015,  2436,  1015,   70,   1;
  1, 169, 5915, 34307, 34307, 5915, 169, 1;
  ...

Alois P. Heinz, Rows n = 0..56, flattened
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
B. Sagan and C. Savage, Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences, arXiv:0911.3159 [math.CO], 2009.
B. Sagan and C. Savage, Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences, Integers 10 (2010), 697-703, A52.

Columns include A000129, A084158, A099930, A099931, A383719.
Row sums are in A099928. Central column is in A099929.
Cf. A010048, A256832, A383715.

p:= proc(n) p(n):= `if`(n<2, n, 2*p(n-1)+p(n-2)) end:
f:= proc(n) f(n):= `if`(n=0, 1, p(n)*f(n-1)) end:
T:= (n, k)-> f(n)/(f(k)*f(n-k)):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Aug 15 2013

p[n_] := p[n] = If[n<2, n, 2*p[n-1] + p[n-2]]; f[n_] := f[n] = If[n == 0, 1, p[n] * f[n-1]]; T[n_, k_] := f[n]/(f[k]*f[n-k]); Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

P(k, n) = Prod[i=k-n+1..k, Pell(i)] / Prod[i=1..n, Pell(i)], with Pell(n) = A000129(n).
From Peter Bala, Mar 15 2013: (Start)
In terms of the Pell numbers, Pell(n) = A000129(n), the triangle entry T(n,k) = [n]!/([k]!*[n-k]!), where [n]! := Pell(1)*...*Pell(n) for n >= 1, with the convention [0]! = 1.
Define E(x) = 1 + sum {n>=0} x^n/[n]!. Then a generating function for this triangle is E(z)*E(x*z) = 1 + (1 + x)*z + (1 + 2*x + x^2)*z^2/[2]! + (1 + 5*x + 5*x^2 + x^3)*z^3/[3]! + ... (End)
G.f. of column k: x^k * exp( Sum_{j>=1} Pell((k+1)*j)/Pell(j) * x^j/j ). - Seiichi Manyama, May 07 2025

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A055870 Signed Fibonomial triangle.

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