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%I #36 Feb 25 2020 16:30:26
%S 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 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 9,43,143,364,728,1093,987,256,1,0,1,10,53,196,560,1288,2380,3280,2584,512,1,0
%N 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 Alois P. Heinz, <a href="/A259475/b259475.txt">Antidiagonals n = 0..140, flattened</a>
%H 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 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 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 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 The first few antidiagonals are:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 2, 1, 0;
%e 1, 3, 4, 1, 0;
%e 1, 4, 8, 8, 1, 0;
%e 1, 5, 13, 21, 16, 1, 0;
%e 1, 6, 19, 40, 55, 32, 1, 0;
%e 1, 7, 26, 66, 121, 144, 64, 1, 0;
%e ...
%e Square array starts:
%e [0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e [1] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e [2] 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
%e [3] 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711, ...
%e [4] 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, 88573, ...
%e [5] 1, 5, 19, 66, 221, 728, 2380, 7753, 25213, 81927, 266110, ...
%e [6] 1, 6, 26, 100, 364, 1288, 4488, 15504, 53296, 182688, 625184, ...
%e [7] 1, 7, 34, 143, 560, 2108, 7752, 28101, 100947, 360526, 1282735, ...
%e [8] 1, 8, 43, 196, 820, 3264, 12597, 47652, 177859, 657800, 2417416, ...
%e [9] 1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, ...
%p F:= proc(n) option remember;
%p `if`(n<2, 1, expand(F(n-1)-t*F(n-2)))
%p end:
%p A:= (n, k)-> coeff(series(1/F(n+1), t, k+1), t, k):
%p seq(seq(A(d-k, k), k=0..d), d=0..12); # _Alois P. Heinz_, Jul 04 2015
%t 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 (SageMath)
%o @cached_function
%o def F(n, k):
%o if k < 0: return 0
%o if k == 0: return 1
%o return sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
%o def A(n, k): return F(n+1, 2*k)
%o print([A(n-k, k) for n in (0..11) for k in (0..n)]) # _Peter Luschny_, Aug 21 2019
%Y The initial rows of the array are A000007, A000012, A000079, A001906, A003432, A005021, A094811, A094256.
%Y A(n,n) gives A274969.
%Y Cf. A309896.
%K nonn,tabl,easy
%O 0,8
%A _N. J. A. Sloane_, Jul 03 2015
%E More terms from _Alois P. Heinz_, Jul 04 2015
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