A259475 Array read by antidiagonals: row n gives coefficients of Taylor series expansion of 1/F_{n+1}(t), where F_i(t) is a Fibonacci polynomial defined by F_0=1, F_1=1, F_{i+1} = F_i-t*F_{i-1}.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 8, 1, 0, 1, 5, 13, 21, 16, 1, 0, 1, 6, 19, 40, 55, 32, 1, 0, 1, 7, 26, 66, 121, 144, 64, 1, 0, 1, 8, 34, 100, 221, 364, 377, 128, 1, 0, 1, 9, 43, 143, 364, 728, 1093, 987, 256, 1, 0, 1, 10, 53, 196, 560, 1288, 2380, 3280, 2584, 512, 1, 0

Offset

0,8

Comments

Table 3.1 in Hopkins thesis is the same below the main diagonal. - F. Chapoton, Sep 04 2025

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Noureddine Chair, Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP¹ into G_r(Cⁿ), arXiv:hep-th/9808170, 1998. (Cf. Table 4).

E. Hopkins, Free Complexes over the Exterior Algebra with Small Homology, Ph.D. thesis, University of Nebraska, 2021. (Cf. Table 3.1.)

G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.

G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]

Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.

Formula

Let F(n, k) = Sum_{j=0..(n-2)/2} (-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for k > 0; F(n, 0) = 1 and F(n, k) = 0 if k < 0. Then A(n, k) = F(n+1, 2*k). See [Shibukawa] and A309896. - Peter Luschny, Aug 21 2019

For n >= k >= 2, A(n, k) = binomial(n+2*k,k-2)*(n^2+3*n-2*k+2)/(k*(k-1)). - F. Chapoton, Sep 05 2025

Example

The first few antidiagonals are:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  1,  0;
  1, 3,  4,  1,   0;
  1, 4,  8,  8,   1,   0;
  1, 5, 13, 21,  16,   1,  0;
  1, 6, 19, 40,  55,  32,  1, 0;
  1, 7, 26, 66, 121, 144, 64, 1, 0;
  ...
Square array starts:
  [0] 1, 0,  0,   0,    0,    0,     0,     0,      0,       0,       0, ...
  [1] 1, 1,  1,   1,    1,    1,     1,     1,      1,       1,       1, ...
  [2] 1, 2,  4,   8,   16,   32,    64,   128,    256,     512,    1024, ...
  [3] 1, 3,  8,  21,   55,  144,   377,   987,   2584,    6765,   17711, ...
  [4] 1, 4, 13,  40,  121,  364,  1093,  3280,   9841,   29524,   88573, ...
  [5] 1, 5, 19,  66,  221,  728,  2380,  7753,  25213,   81927,  266110, ...
  [6] 1, 6, 26, 100,  364, 1288,  4488, 15504,  53296,  182688,  625184, ...
  [7] 1, 7, 34, 143,  560, 2108,  7752, 28101, 100947,  360526, 1282735, ...
  [8] 1, 8, 43, 196,  820, 3264, 12597, 47652, 177859,  657800, 2417416, ...
  [9] 1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, ...

Maple

F:= proc(n) option remember;
      `if`(n<2, 1, expand(F(n-1)-t*F(n-2)))
    end:
A:= (n, k)-> coeff(series(1/F(n+1), t, k+1), t, k):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 04 2015

Mathematica

F[n_] := F[n] = If[n<2, 1, Expand[F[n-1] - t*F[n-2]]]; A[n_, k_] := SeriesCoefficient[1/F[n+1], { t, 0, k}]; Table[A[d-k, k], {d, 0, 12}, {k, 0, d}] // Flatten (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)

Prog

(SageMath)
@cached_function
def F(n, k):
    if k <  0: return 0
    if k == 0: return 1
    return sum((-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for j in (0..(n-2)/2))
def A(n, k): return F(n+1, 2*k)
print([A(n-k, k) for n in (0..11) for k in (0..n)]) # Peter Luschny, Aug 21 2019
(Python)  # The lower triangular array computed by F. Chapoton's formula:
from math import comb as binomial
def T(n: int, k: int) -> int:
    if k < 0:  return 0
    if k == 0: return 1
    if k == 1: return n
    return (binomial(n + 2*k, k - 2) * (n**2 + 3*n - 2*k + 2)) // (k * (k - 1))
for n in range(9): print([T(n, k) for k in range(n+1)])  # Peter Luschny, Sep 06 2025

Crossrefs

The initial rows of the array are A000007, A000012, A000079, A001906, A003432, A005021, A094811, A094256.

A(n,n) gives A274969.

Cf. A309896.

A188843 is a variant without the first two rows and the first column, and the antidiagonals read in opposite direction.

Sequence in context: A286932 A350364 A358575 * A361952 A323224 A118340

Adjacent sequences: A259472 A259473 A259474 * A259476 A259477 A259478

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jul 03 2015

Extensions

More terms from Alois P. Heinz, Jul 04 2015

Status

approved