A304359 Antidiagonal sums of the second quadrant of array A(k,m) = F_k(m), F_k(m) being the k-th Fibonacci polynomial evaluated at m.

0, 1, 1, 1, 1, 1, 0, 2, 1, -10, 39, -58, -166, 1611, -6311, 10083, 54195, -565257, 2727568, -6102368, -26464605, 394614352, -2515452801, 8797315672, 11441288836, -458369484247, 4097437715969, -21769011878335, 36715605929957, 703213495381553, -10042075731879152

Offset: 0

Author

Alois P. Heinz, May 11 2018

Keywords

• sign

Comments

Equivalently, antidiagonal sums of the fourth quadrant of array A(k,m).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..616
• Wikipedia, Fibonacci polynomials
• Wikipedia, Quadrant (plane geometry)

Crossrefs

Cf. A000045, A084844, A304357.

Programs

Maple

F:= (n, k)-> (<<0|1>, <1|k>>^n)[1, 2]:
a:= n-> add(F(-j, n-j), j=0..n):
seq(a(n), n=0..30);
# second Maple program:
F:= proc(n, k) option remember;
      `if`(n<2, n, k*F(n-1, k)+F(n-2, k))
    end:
a:= n-> add(F(j, j-n), j=0..n):
seq(a(n), n=0..30);
# third Maple program:
a:= n-> add(combinat[fibonacci](j, j-n), j=0..n):
seq(a(n), n=0..30);

Mathematica

a[n_] := Sum[Fibonacci[j, j - n], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 02 2018, from 3rd Maple program *)

Formula

a(n) = Sum_{j=0..n} F_j(j-n).