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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}.

%I A259475 #49 Sep 03 2025 15:32:29
%S A259475 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,
%T A259475 55,32,1,0,1,7,26,66,121,144,64,1,0,1,8,34,100,221,364,377,128,1,0,1,
%U A259475 9,43,143,364,728,1093,987,256,1,0,1,10,53,196,560,1288,2380,3280,2584,512,1,0
%N 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}.
%H A259475 Alois P. Heinz, <a href="/A259475/b259475.txt">Antidiagonals n = 0..140, flattened</a>
%H A259475 Noureddine Chair, <a href="https://arxiv.org/pdf/hep-th/9808170">Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP¹ into G<sub>r</sub>(C<sup>n</sup>)</a>, arXiv:hep-th/9808170, 1998. (Cf. Table 4).
%H A259475 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1970__15__3_0">Sur les éventails de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
%H A259475 G. Kreweras, <a href="/A000108/a000108_1.pdf">Sur les éventails de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
%H A259475 Genki Shibukawa, <a href="https://arxiv.org/abs/1907.00334">New identities for some symmetric polynomials and their applications</a>, arXiv:1907.00334 [math.CA], 2019.
%F A259475 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
%e A259475 The first few antidiagonals are:
%e A259475   1;
%e A259475   1, 0;
%e A259475   1, 1,  0;
%e A259475   1, 2,  1,  0;
%e A259475   1, 3,  4,  1,   0;
%e A259475   1, 4,  8,  8,   1,   0;
%e A259475   1, 5, 13, 21,  16,   1,  0;
%e A259475   1, 6, 19, 40,  55,  32,  1, 0;
%e A259475   1, 7, 26, 66, 121, 144, 64, 1, 0;
%e A259475   ...
%e A259475 Square array starts:
%e A259475   [0] 1, 0,  0,   0,    0,    0,     0,     0,      0,       0,       0, ...
%e A259475   [1] 1, 1,  1,   1,    1,    1,     1,     1,      1,       1,       1, ...
%e A259475   [2] 1, 2,  4,   8,   16,   32,    64,   128,    256,     512,    1024, ...
%e A259475   [3] 1, 3,  8,  21,   55,  144,   377,   987,   2584,    6765,   17711, ...
%e A259475   [4] 1, 4, 13,  40,  121,  364,  1093,  3280,   9841,   29524,   88573, ...
%e A259475   [5] 1, 5, 19,  66,  221,  728,  2380,  7753,  25213,   81927,  266110, ...
%e A259475   [6] 1, 6, 26, 100,  364, 1288,  4488, 15504,  53296,  182688,  625184, ...
%e A259475   [7] 1, 7, 34, 143,  560, 2108,  7752, 28101, 100947,  360526, 1282735, ...
%e A259475   [8] 1, 8, 43, 196,  820, 3264, 12597, 47652, 177859,  657800, 2417416, ...
%e A259475   [9] 1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, ...
%p A259475 F:= proc(n) option remember;
%p A259475       `if`(n<2, 1, expand(F(n-1)-t*F(n-2)))
%p A259475     end:
%p A259475 A:= (n, k)-> coeff(series(1/F(n+1), t, k+1), t, k):
%p A259475 seq(seq(A(d-k, k), k=0..d), d=0..12);  # _Alois P. Heinz_, Jul 04 2015
%t A259475 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_ *)
%o A259475 (SageMath)
%o A259475 @cached_function
%o A259475 def F(n, k):
%o A259475     if k <  0: return 0
%o A259475     if k == 0: return 1
%o A259475     return sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
%o A259475 def A(n, k): return F(n+1, 2*k)
%o A259475 print([A(n-k, k) for n in (0..11) for k in (0..n)]) # _Peter Luschny_, Aug 21 2019
%Y A259475 The initial rows of the array are A000007, A000012, A000079, A001906, A003432, A005021, A094811, A094256.
%Y A259475 A(n,n) gives A274969.
%Y A259475 Cf. A309896.
%Y A259475 A188843 is a variant without the first two rows and the first column, and the antidiagonals read in opposite direction.
%K A259475 nonn,tabl,easy,changed
%O A259475 0,8
%A A259475 _N. J. A. Sloane_, Jul 03 2015
%E A259475 More terms from _Alois P. Heinz_, Jul 04 2015