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%I #35 Jun 23 2023 17:00:25
%S 1,1,2,1,3,1,1,4,5,-1,1,5,11,7,-2,1,6,19,29,9,-1,1,7,29,71,76,11,1,1,
%T 8,41,139,265,199,13,2,1,9,55,239,666,989,521,15,1,1,10,71,377,1393,
%U 3191,3691,1364,17,-1,1,11,89,559,2584,8119,15289,13775,3571,19,-2
%N Rectangular array read by (upward) antidiagonals: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)*binomial(k-floor((j+1)/2), floor(j/2))*n^(k-j), n >= 1, k >= 0.
%C This array is used to compute A269254: A269254(n) = least k such that A(n,k) is a prime, or -1 if no such k exists.
%C For detailed theory, see [Hone]. - _L. Edson Jeffery_, Feb 09 2018
%C The array can be extended to k<0 with A(n, k) = -A(n, -k-1) for all k in Z. - _Michael Somos_, Jun 19 2023
%H Andrew N. W. Hone, et al., <a href="https://arxiv.org/abs/1802.01793">On a family of sequences related to Chebyshev polynomials</a>, arXiv:1802.01793 [math.NT], 2018.
%F A(n,0) = 1, A(n,1) = n + 1, A(n,k) = n*A(n,k-1) - A(n,k-2), n >= 1, k >= 2.
%F G.f. for row n: (1 + x)/(1 - n*x + x^2), n >= 1.
%F A(n, k) = B(-n, k) where B = A29045. - _Michael Somos_, Jun 19 2023
%e Array begins:
%e 1 2 1 -1 -2 -1 1 2 1 -1
%e 1 3 5 7 9 11 13 15 17 19
%e 1 4 11 29 76 199 521 1364 3571 9349
%e 1 5 19 71 265 989 3691 13775 51409 191861
%e 1 6 29 139 666 3191 15289 73254 350981 1681651
%e 1 7 41 239 1393 8119 47321 275807 1607521 9369319
%e 1 8 55 377 2584 17711 121393 832040 5702887 39088169
%e 1 9 71 559 4401 34649 272791 2147679 16908641 133121449
%e 1 10 89 791 7030 62479 555281 4935050 43860169 389806471
%e 1 11 109 1079 10681 105731 1046629 10360559 102558961 1015229051
%t (* Array: *)
%t Grid[Table[LinearRecurrence[{n, -1}, {1, 1 + n}, 10], {n, 10}]]
%t (* Array antidiagonals flattened (gives this sequence): *)
%t A294099[n_, k_] := Sum[(-1)^(Floor[j/2]) Binomial[k - Floor[(j + 1)/2], Floor[j/2]] n^(k - j), {j, 0, k}]; Flatten[Table[A294099[n - k, k], {n, 11}, {k, 0, n - 1}]]
%o (PARI) {A(n, k) = sum(j=0, k, (-1)^(j\2)*binomial(k-(j+1)\2, j\2)*n^(k-j))}; /* _Michael Somos_, Jun 19 2023 */
%Y Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
%Y Rows: A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, ...
%Y Columns: A000012, A000027, A028387, ...
%K sign,tabl
%O 1,3
%A _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Oct 22 2017
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