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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.

%I A304359 #18 Jun 02 2018 14:18:35
%S A304359 0,1,1,1,1,1,0,2,1,-10,39,-58,-166,1611,-6311,10083,54195,-565257,
%T A304359 2727568,-6102368,-26464605,394614352,-2515452801,8797315672,
%U A304359 11441288836,-458369484247,4097437715969,-21769011878335,36715605929957,703213495381553,-10042075731879152
%N 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.
%C A304359 Equivalently, antidiagonal sums of the fourth quadrant of array A(k,m).
%H A304359 Alois P. Heinz, <a href="/A304359/b304359.txt">Table of n, a(n) for n = 0..616</a>
%H A304359 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_polynomials">Fibonacci polynomials</a>
%H A304359 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quadrant_(plane_geometry)">Quadrant (plane geometry)</a>
%F A304359 a(n) = Sum_{j=0..n} F_j(j-n).
%p A304359 F:= (n, k)-> (<<0|1>, <1|k>>^n)[1, 2]:
%p A304359 a:= n-> add(F(-j, n-j), j=0..n):
%p A304359 seq(a(n), n=0..30);
%p A304359 # second Maple program:
%p A304359 F:= proc(n, k) option remember;
%p A304359       `if`(n<2, n, k*F(n-1, k)+F(n-2, k))
%p A304359     end:
%p A304359 a:= n-> add(F(j, j-n), j=0..n):
%p A304359 seq(a(n), n=0..30);
%p A304359 # third Maple program:
%p A304359 a:= n-> add(combinat[fibonacci](j, j-n), j=0..n):
%p A304359 seq(a(n), n=0..30);
%t A304359 a[n_] := Sum[Fibonacci[j, j - n], {j, 0, n}];
%t A304359 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 02 2018, from 3rd Maple program *)
%Y A304359 Cf. A000045, A084844, A304357.
%K A304359 sign
%O A304359 0,8
%A A304359 _Alois P. Heinz_, May 11 2018