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.

0, 1, 1, 3, 5, 13, 32, 94, 297, 1036, 3911, 15918, 69350, 321779, 1582745, 8220349, 44925187, 257563819, 1544896976, 9671289892, 63051738167, 427254561854, 3003872526303, 21876513464296, 164790822258172, 1282198404741305, 10292007232817249, 85126350266370355

Offset

0,4

Comments

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

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..600

Wikipedia, Fibonacci polynomials

Wikipedia, Quadrant (plane geometry)

Formula

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

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

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, n-j), j=0..n):
seq(a(n), n=0..30);
# third Maple program:
a:= n-> add(combinat[fibonacci](j, n-j), j=0..n):
seq(a(n), n=0..30);

Mathematica

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

Crossrefs

Cf. A000045, A084844, A304359, A101494, A172236, A157103.

Sequence in context: A127600 A262237 A051401 * A296023 A027039 A283844

Adjacent sequences: A304354 A304355 A304356 * A304358 A304359 A304360

Keyword

nonn

Author

Alois P. Heinz, May 11 2018

Status

approved