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 A055870 Signed Fibonomial triangle. 27

%I

%S 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 -8,-1,1,-13,-104,260,260,-104,-13,1,1,-21,-273,1092,1820,-1092,-273,

%U 21,1,1,-34,-714,4641,12376,-12376,-4641,714,34,-1,1,-55,-1870,19635,85085,-136136,-85085,19635,1870,-55,-1

%N Signed Fibonomial triangle.

%C 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.

%C 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.

%C 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).

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

%H 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 A. Brousseau, <a href="http://www.fq.math.ca/Scanned/6-1/brousseau3.pdf">A sequence of power formulas</a>, Fib. Quart., 6 (1968), 81-83.

%H H. W. Gould, <a href="http://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 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 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/Fibonomials.html">The Fibonomials</a>

%H 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 A. K. Kwasniewski, <a href="http://arXiv.org/abs/math/0406006">Fibonomial cumulative connection constants</a>, arXiv:math/0406006 [math.CO], 2004-2009.

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

%F 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).

%e 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 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.

%e The triangle begins:

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

%e 0 1

%e 1 1 -1

%e 2 1 -1 -1

%e 3 1 -2 -2 1

%e 4 1 -3 -6 3 1

%e 5 1 -5 -15 15 5 -1

%e 6 1 -8 -40 60 40 -8 -1

%e 7 1 -13 -104 260 260 -104 -13 1

%e 8 1 -21 -273 1092 1820 -1092 -273 21 1

%e 9 1 -34 -714 4641 12376 -12376 -4641 714 34 -1

%e ... [_Wolfdieter Lang_, Aug 06 2012; a(7,1) corrected, Oct 10 2012]

%p A055870 := proc(n,k)

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

%p end proc: # _R. J. Mathar_, Jun 14 2015

%t 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 *)

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

%Y Cf. A051159.

%Y Central column: A003268.

%K easy,sign,tabl

%O 0,8

%A _Wolfdieter Lang_, Jul 10 2000

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Last modified February 22 14:19 EST 2020. Contains 332136 sequences. (Running on oeis4.)