0,5

G.f. for column m: see column sequences: A000012, A000071, A056589-91, for m=0..4.

The row polynomials p(n,x) := sum(a(n,m)*x^m) occur as numerators of the g.f. for the (n+1)-th power of Fibonacci numbers A000045. The corresponding denominator polynomials are the row polynomials q(n+2,x)= sum(A055870(n+2,m)*x^m,m=0..n+2) (signed Fibonomial triangle).

The row polynomials p(n,x) and the companion denominator polynomials q(n,x) can be deduced from Riordan's recursion result.

The explicit formula is found from the recursion relation for powers of Fibonacci numbers (see Knuth's exercise with solution).

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, p. 84, (exercise 1.2.8. Nr. 30) and p. 492 (solution).

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

a(n, m)=0 if n<m; a(n, 0)=1; a(n, m)= (F(m+1))^(n+1)+sum(sfibonomial(n+2, j)*(F(m+1-j)^(n+1)), j=1..m) m=1..n, with F(n)=A000045(n) (Fibonacci) and sfibonomial(n, m) := A055870(n, m) (signed Fibonomial triangle).

Row polynomial for n=4: p(4,x)=1-7*x-16*x^2+7*x^3+x^4. x*p(4,x) is the numerator of the g.f. for A056572(n), n >= 0 (fifth power of Fibonacci numbers) {0,1,1,32,243,...}. The denominator polynomial is sum(A055870(6,m)*x^m,m=0..6) (n=6 row polynomial of signed fibonomial triangle).

A055870, A000012, A000071, A056589-91, A000045, A007598, A056570-4, A056585-7.

Sequence in context: A104382 A086629 A126770 this_sequence A137854 A062715 A100631

Adjacent sequences: A056585 A056586 A056587 this_sequence A056589 A056590 A056591

easy,sign,tabl

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