A092879 Triangle of coefficients of the product of two consecutive Fibonacci polynomials.

1, 1, 1, 1, 3, 2, 1, 5, 7, 2, 1, 7, 16, 13, 3, 1, 9, 29, 40, 22, 3, 1, 11, 46, 91, 86, 34, 4, 1, 13, 67, 174, 239, 166, 50, 4, 1, 15, 92, 297, 541, 553, 296, 70, 5, 1, 17, 121, 468, 1068, 1461, 1163, 496, 95, 5, 1, 19, 154, 695, 1912, 3300, 3544, 2269, 791, 125, 6, 1, 21, 191

Offset

0,5

Comments

The Fibonacci polynomials are defined by F(0,x) = 1, F(1,x) = 1 and F(n, x) = F(n-1, x) + x*F(n-2, x).

This is also the reflected triangle of coefficients of the polynomials defined by the recursion: c0=-1; p(x, n) = (2 + c0 - x)*p(x, n - 1) + (-1 - c0*(2 - x))*p(x, n - 2) + c0*p(x, n - 3). - Roger L. Bagula, Apr 09 2008

Links

Table of n, a(n) for n=0..68.

Formula

From Johannes W. Meijer, Jul 20 2011: (Start)

T(n, k) = Sum_{i=0..k} (-1)^(i+k)*binomial(i+2*n-2*k+1, i).

T(n, k) = A035317(2*n-k, k) = A158909(n, n-k.) (End)

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) - T(n-3,k-3), T(0,0) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 12 2013

Example

Triangle begins;
  1;
  1,1;
  1,3,2;
  1,5,7,2;
  1,7,16,13,3;
  1,9,29,40,22,3;
  ...
F(3,x) = 1 + 2*x and F(4,x) = 1 + 3*x + x^2 so F(3,x)*F(4,x)=(1 + 3*x + x^2)*(1 + 2*x) = 1 + 5*x + 7*x^2 + 2*x^3 leads to T(3,k) = [1,5,7,2].

Maple

T:=proc(n, k): add((-1)^(i+k)*binomial(i+2*n-2*k+1, i), i=0..k) end: seq(seq(T(n, k), k=0..n), n=0..10); # Johannes W. Meijer, Jul 20 2011
T:=proc(n, k): coeff(F(n, x)*F(n+1, x), x, k) end: F:=proc(n, x) option remember: if n=0 then 1 elif n=1 then 1 else procname(n-1, x) + x*procname(n-2, x) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # Johannes W. Meijer, Jul 20 2011

Mathematica

c0 = -1; p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = 2 - x + c0; p[x_, n_] :=p[x, n] = (2 + c0 -x)*p[x, n - 1] + (-1 - c0 (2 - x))*p[x, n - 2] + c0*p[x, n - 3]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[Reverse[CoefficientList[p[x, n], x]], {n, 0, 10}]; Flatten[a] (* Roger L. Bagula, Apr 09 2008 *)

Prog

(PARI) T(n, k)=local(m); if(k<0 || k>n, 0, n++; m=contfracpnqn(matrix(2, n, i, j, x)); polcoeff(m[1, 1]*m[2, 1]/x^n, n-k))

Crossrefs

Row sums are A001654(n+1).

Cf. A109954, A136674.

Sequence in context: A110712 A065366 A360364 * A073370 A208511 A129675

Adjacent sequences: A092876 A092877 A092878 * A092880 A092881 A092882

Keyword

nonn,tabl

Author

Michael Somos, Mar 10 2004

Extensions

Edited and information added by Johannes W. Meijer, Jul 20 2011

Status

approved