This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A299045 #40 Jul 19 2024 08:46:46
%S A299045 1,1,0,1,-1,-1,1,-2,1,1,1,-3,5,-1,0,1,-4,11,-13,1,-1,1,-5,19,-41,34,
%T A299045 -1,1,1,-6,29,-91,153,-89,1,0,1,-7,41,-169,436,-571,233,-1,-1,1,-8,55,
%U A299045 -281,985,-2089,2131,-610,1,1,1,-9,71,-433,1926,-5741,10009,-7953,1597,-1,0
%N A299045 Rectangular array: 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, read by antidiagonals.
%C A299045 This array is used to compute A269252: A269252(n) = least k such that |A(n,k)| is a prime, or -1 if no such k exists.
%C A299045 For detailed theory, see [Hone].
%C A299045 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 A299045 Robert Price, <a href="/A299045/b299045.txt">Table of n, a(n) for n = 1..5050</a>
%H A299045 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 A299045 G.f. for row n: (1 + x)/(1 + n*x + x^2), n >= 1.
%F A299045 A(n, k) = B(-n, k) where B = A294099. - _Michael Somos_, Jun 19 2023
%e A299045 Array begins:
%e A299045 1 0 -1 1 0 -1 1 0 -1 1
%e A299045 1 -1 1 -1 1 -1 1 -1 1 -1
%e A299045 1 -2 5 -13 34 -89 233 -610 1597 -4181
%e A299045 1 -3 11 -41 153 -571 2131 -7953 29681 -110771
%e A299045 1 -4 19 -91 436 -2089 10009 -47956 229771 -1100899
%e A299045 1 -5 29 -169 985 -5741 33461 -195025 1136689 -6625109
%e A299045 1 -6 41 -281 1926 -13201 90481 -620166 4250681 -29134601
%e A299045 1 -7 55 -433 3409 -26839 211303 -1663585 13097377 -103115431
%e A299045 1 -8 71 -631 5608 -49841 442961 -3936808 34988311 -310957991
%e A299045 1 -9 89 -881 8721 -86329 854569 -8459361 83739041 -828931049
%t A299045 (* Array: *)
%t A299045 Grid[Table[LinearRecurrence[{-n, -1}, {1, 1 - n}, 10], {n, 10}]]
%t A299045 (*Array antidiagonals flattened (gives this sequence):*)
%t A299045 A299045[n_, k_] := Sum[(-1)^(Floor[j/2]) Binomial[k - Floor[(j + 1)/2], Floor[j/2]] (-n)^(k - j), {j, 0, k}]; Flatten[Table[A299045[n - k, k], {n, 11}, {k, 0, n - 1}]]
%o A299045 (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 A299045 Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269251, A269252, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
%Y A299045 Cf. A094954 (unsigned version of this array, but missing the first row).
%Y A299045 Cf. Rows: A057078, A033999, A099496, A079935 (or A001835), A004253, A001653, A049685, A070997, A070998, A138288 (or A072256), ...
%Y A299045 Cf. Columns: A000012, A001477 (A000027), A110331 (A165900), A123972, A192398, ...
%K A299045 sign,tabl
%O A299045 1,8
%A A299045 _L. Edson Jeffery_, _Bob Selcoe_ and _Andrew Hone_, Feb 01 2018