0,8

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 polynomials 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-x, p(1,x)= 1-x-x^2 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 (pbarry(AT)wit.ie), Oct 05 2004

A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.

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

J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.

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).

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.

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

Cf. A051159.

Central column: A003268.

Adjacent sequences: A055867 A055868 A055869 this_sequence A055871 A055872 A055873

Sequence in context: A131791 A010358 A010048 this_sequence A136512 A088459 A007799

easy,sign,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jul 10 2000