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
Examples
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, ...
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 Gr(Cn), arXiv:hep-th/9808170, 1998. (Cf. Table 4).
- 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.
Crossrefs
Programs
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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 *) -
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
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
Extensions
More terms from Alois P. Heinz, Jul 04 2015
Comments