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

Links

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

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

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

Crossrefs

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

A(n,n) gives A274969.

Cf. A309896.

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