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A304357 Antidiagonal sums of the first quadrant of array A(k,m) = F_k(m), F_k(m) being the k-th Fibonacci polynomial evaluated at m.

%I A304357 #23 Mar 24 2021 03:21:36
%S A304357 0,1,1,3,5,13,32,94,297,1036,3911,15918,69350,321779,1582745,8220349,
%T A304357 44925187,257563819,1544896976,9671289892,63051738167,427254561854,
%U A304357 3003872526303,21876513464296,164790822258172,1282198404741305,10292007232817249,85126350266370355
%N A304357 Antidiagonal sums of the first quadrant of array A(k,m) = F_k(m), F_k(m) being the k-th Fibonacci polynomial evaluated at m.
%C A304357 Equivalently, antidiagonal sums of the third quadrant of array A(k,m).
%C A304357 It seems that: a(n+1) is the sum of the n-th antidiagonal of triangle A101494; a(n)-(n mod 2) is the sum of the n-th antidiagonal of array A172236; and a(n+1)+(n mod 2) is the sum of row n of triangle A157103. - _Mathew Englander_, Feb 28 2021
%H A304357 Alois P. Heinz, <a href="/A304357/b304357.txt">Table of n, a(n) for n = 0..600</a>
%H A304357 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_polynomials">Fibonacci polynomials</a>
%H A304357 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quadrant_(plane_geometry)">Quadrant (plane geometry)</a>
%F A304357 a(n) = Sum_{j=0..n} F_j(n-j).
%F A304357 a(n+1) = Sum_{j = 0..n} Sum_{i = j..floor((n+j)/2)} binomial(i,j)*(n+j-2*i)^j (empirically). - _Mathew Englander_, Feb 28 2021
%p A304357 F:= (n, k)-> (<<0|1>, <1|k>>^n)[1, 2]:
%p A304357 a:= n-> add(F(j, n-j), j=0..n):
%p A304357 seq(a(n), n=0..30);
%p A304357 # second Maple program:
%p A304357 F:= proc(n, k) option remember;
%p A304357       `if`(n<2, n, k*F(n-1, k)+F(n-2, k))
%p A304357     end:
%p A304357 a:= n-> add(F(j, n-j), j=0..n):
%p A304357 seq(a(n), n=0..30);
%p A304357 # third Maple program:
%p A304357 a:= n-> add(combinat[fibonacci](j, n-j), j=0..n):
%p A304357 seq(a(n), n=0..30);
%t A304357 a[n_] := Sum[Fibonacci[j, n - j], {j, 0, n}];
%t A304357 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 02 2018, from 3rd Maple program *)
%Y A304357 Cf. A000045, A084844, A304359, A101494, A172236, A157103.
%K A304357 nonn
%O A304357 0,4
%A A304357 _Alois P. Heinz_, May 11 2018