cp's OEIS Frontend

A056588 Coefficient triangle of certain polynomials.

%I A056588 #40 Jan 05 2025 19:51:36
%S A056588 1,1,-1,1,-2,-1,1,-4,-4,1,1,-7,-16,7,1,1,-12,-53,53,12,-1,1,-20,-166,
%T A056588 318,166,-20,-1,1,-33,-492,1784,1784,-492,-33,1,1,-54,-1413,9288,
%U A056588 17840,-9288,-1413,54,1,1,-88,-3960,46233,163504,-163504,-46233,3960,88,-1
%N A056588 Coefficient triangle of certain polynomials.
%C A056588 G.f. for column m: see column sequences: A000012, A000071, A056589-91, for m=0..4.
%C A056588 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_{m=0..n+2} A055870(n+2, m)*x^m (signed Fibonomial triangle).
%C A056588 The row polynomials p(n,x) and the companion denominator polynomials q(n,x) can be deduced from Riordan's recursion result.
%C A056588 The explicit formula is found from the recursion relation for powers of Fibonacci numbers (see Knuth's exercise with solution). - _Roger L. Bagula_, Apr 03 2010
%D A056588 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).
%H A056588 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 A056588 S. Falcon, <a href="http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
%H A056588 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.
%F A056588 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).
%F A056588 From _Roger L. Bagula_, Apr 03 2010: (Start)
%F A056588 p(x,n) = Sum_{k>=0} (((1 + sqrt(5))^k - (1 - sqrt(5))^k)/(2^k*sqrt(5)))^n*x^k;
%F A056588 t(n,m) = Numerator_coefficients(p(x,n)/x)/2^(1 + floor(n/2));
%F A056588 out(n,m) = t(n,m)/t(n,1). (End)
%F A056588 T(n, k) = Sum_{j=0..k} Fibonacci(k+1-j)^(n+1) * A055870(n+2, j). - _Tony Foster III_, Aug 20 2018
%F A056588 Sum_{j=0..n-1} a(n-1, n-1-j)*A010048(k+j, n) = Fibonacci(k)^n. - _Tony Foster III_, Jul 24 2018
%e A056588 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_{m=0..6} A055870(6,m)*x^m (n=6 row polynomial of signed fibonomial triangle).
%e A056588 From _Roger L. Bagula_, Apr 03 2010: (Start)
%e A056588 1;
%e A056588 1,  -1;
%e A056588 1,  -2,    -1;
%e A056588 1,  -4,    -4,     1;
%e A056588 1,  -7,   -16,     7,      1;
%e A056588 1, -12,   -53,    53,     12,      -1;
%e A056588 1, -20,  -166,   318,    166,     -20,     -1;
%e A056588 1, -33,  -492,  1784,   1784,    -492,    -33,    1;
%e A056588 1, -54, -1413,  9288,  17840,   -9288,  -1413,   54,  1;
%e A056588 1, -88, -3960, 46233, 163504, -163504, -46233, 3960, 88, -1; (End)
%p A056588 A056588 := proc(n,k)
%p A056588     if k = 0 then
%p A056588         1;
%p A056588     elif k >n then
%p A056588         0;
%p A056588     else
%p A056588         combinat[fibonacci](k+1)^(n+1)+add( A055870(n+2, j)*(combinat[fibonacci](k+1-j)^(n+1)), j=1..k) ;
%p A056588     end if;
%p A056588 end proc: # _R. J. Mathar_, Jun 14 2015
%t A056588 p[x_, n_] = Sum[(((1 + Sqrt[5])^k - (1 - Sqrt[5])^k)/(2^k*Sqrt[5]))^n*x^k, {k, 0, Infinity}];
%t A056588 a = Table[CoefficientList[FullSimplify[Numerator[p[ x, n]]/x], x]/2^(1 + Floor[n/2]), {n, 1, 10}];
%t A056588 Table[a[[n]]/a[[n]][[1]], {n, 1, 10}];
%t A056588 Flatten[%] (* _Roger L. Bagula_, Apr 03 2010 *)
%o A056588 (PARI) S(n, k) = (-1)^floor((k+1)/2)*(prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)));
%o A056588 T(n, k) = sum(j=0, k, fibonacci(k+1-j)^(n+1) * S(n+2, j));
%o A056588 tabl(m) = for (n=0, m, for (k=0, n, print1(T(n, k), ", ")); print);
%o A056588 tabl(9); \\ _Tony Foster III_, Aug 20 2018
%Y A056588 Cf. A055870, A000012, A000071, A056589-91, A056592 (row sums), A000045, A007598, A056570-4, A056585-7.
%K A056588 easy,sign,tabl
%O A056588 0,5
%A A056588 _Wolfdieter Lang_, Jul 10 2000