This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A055870 #75 May 07 2025 11:17:09
%S A055870 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,
%T A055870 -8,-1,1,-13,-104,260,260,-104,-13,1,1,-21,-273,1092,1820,-1092,-273,
%U A055870 21,1,1,-34,-714,4641,12376,-12376,-4641,714,34,-1,1,-55,-1870,19635,85085,-136136,-85085,19635,1870,-55,-1
%N A055870 Signed Fibonomial triangle.
%C A055870 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.
%C A055870 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.
%C A055870 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).
%C A055870 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
%D A055870 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, pp. 84-5 and 492.
%H A055870 G. C. Greubel, <a href="/A055870/b055870.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A055870 Katharine A. Ahrens, <a href="https://repository.lib.ncsu.edu/bitstream/handle/1840.20/37364/etd.pdf">Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis</a>, Ph. D. thesis, North Carolina State University (2020).
%H A055870 A. T. Benjamin, S. S. Plott, <a href="http://www.math.hmc.edu/~benjamin/papers/Fibonomial.pdf">A combinatorial approach to fibonomial coefficients</a>, Fib. Quart. 46/47 (1) (2008/9) 7-9.
%H A055870 A. Brousseau, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/6-1/brousseau3.pdf">A sequence of power formulas</a>, Fib. Quart., 6 (1968), 81-83.
%H A055870 H. W. Gould, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/33-5/gould.pdf">Extensions of the Hermite g.c.d. theorems for binomial coefficients</a>, Fib Quart. 33 (1995) 386.
%H A055870 E. Kilic, <a href="http://dx.doi.org/10.1016/j.ejc.2009.03.041">The generalized Fibonomial matrix</a>, Eur. J. Combinat. 31 (1) (2010) 193-209.
%H A055870 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/Fibonomials.html">The Fibonomials</a>
%H A055870 Ewa Krot, <a href="http://dx.doi.org/10.2478/BF02475975">An introduction to finite fibonomial calculus</a>, Centr. Eur. J. Math. 2 (5) (2004) 754.
%H A055870 A. K. Kwasniewski, <a href="http://arXiv.org/abs/math/0406006">Fibonomial cumulative connection constants</a>, arXiv:math/0406006 [math.CO], 2004-2009.
%H A055870 Phakhinkon Phunphayap, <a href="http://ithesis-ir.su.ac.th/dspace/bitstream/123456789/3040/1/59305804.pdf">Various Problems Concerning Factorials, Binomial Coefficients, Fibonomial Coefficients, and Palindromes</a>, Ph. D. Thesis, Silpakorn University (Thailand 2021).
%H A055870 Phakhinkon Phunphayap, Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Pongsriiam/pong12.html">Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients</a>, J. Int. Seq. 21 (2018), #18.3.1.
%H A055870 J. Riordan, <a href="http://dx.doi.org/10.1215/S0012-7094-62-02902-2">Generating functions for powers of Fibonacci numbers</a>, Duke. Math. J. 29 (1962) 5-12.
%H A055870 J. Seibert and P. Trojovsky, <a href="http://dspace.dml.cz/handle/10338.dmlcz/130423">On some identities for the Fibonomial coefficients</a>, Math. Slov. 55 (2005) 9-19.
%H A055870 P. Trojovsky, <a href="http://dx.doi.org/10.1016/j.dam.2007.05.003">On some identities for the Fibonomial coefficients...</a>, Discr. Appl. Math. 155 (15) (2007) 2017.
%F A055870 T(n, m) = (-1)^floor((m+1)/2)*A010048(n, m), where A010048(n, m) := fibonomial(n, m).
%F A055870 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.
%F A055870 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
%e A055870 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).
%e A055870 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.
%e A055870 The triangle begins:
%e A055870 n\m 0 1 2 3 4 5 6 7 8 9
%e A055870 0 1
%e A055870 1 1 -1
%e A055870 2 1 -1 -1
%e A055870 3 1 -2 -2 1
%e A055870 4 1 -3 -6 3 1
%e A055870 5 1 -5 -15 15 5 -1
%e A055870 6 1 -8 -40 60 40 -8 -1
%e A055870 7 1 -13 -104 260 260 -104 -13 1
%e A055870 8 1 -21 -273 1092 1820 -1092 -273 21 1
%e A055870 9 1 -34 -714 4641 12376 -12376 -4641 714 34 -1
%e A055870 ... [_Wolfdieter Lang_, Aug 06 2012; a(7,1) corrected, Oct 10 2012]
%p A055870 A055870 := proc(n,k)
%p A055870 (-1)^floor((k+1)/2)*A010048(n,k) ;
%p A055870 end proc: # _R. J. Mathar_, Jun 14 2015
%t A055870 T[n_, m_]:= {1,-1,-1,1}[[Mod[m,4] + 1]] * Product[ Fibonacci[n-j+1]/Fibonacci[j], {j, m}];
%t A055870 Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* _Jean-François Alcover_, Jul 05 2013 *)
%o A055870 (Magma)
%o A055870 Fibonomial:= func< n,k | k eq 0 select 1 else (&*[Fibonacci(n-j+1)/Fibonacci(j): j in [1..k]]) >;
%o A055870 [(-1)^Floor((k+1)/2)*Fibonomial(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 20 2024
%o A055870 (SageMath)
%o A055870 def fibonomial(n,k): return 1 if k==0 else product(fibonacci(n-j+1)/fibonacci(j) for j in range(1,k+1))
%o A055870 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
%Y A055870 Cf. A000032, A000045, A001654, A001655, A001656, A001657, A001658, A010048, A051159, A056565, A056566, A056567.
%Y A055870 Sums include: A055871 (signed row), A056569 (row).
%Y A055870 Central column: A003268.
%Y A055870 Cf. A383715.
%K A055870 easy,sign,tabl
%O A055870 0,8
%A A055870 _Wolfdieter Lang_, Jul 10 2000