OFFSET
1,8
COMMENTS
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.
For detailed theory, see [Hone].
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
LINKS
Robert Price, Table of n, a(n) for n = 1..5050
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.
FORMULA
G.f. for row n: (1 + x)/(1 + n*x + x^2), n >= 1.
A(n, k) = B(-n, k) where B = A294099. - Michael Somos, Jun 19 2023
EXAMPLE
Array begins:
1 0 -1 1 0 -1 1 0 -1 1
1 -1 1 -1 1 -1 1 -1 1 -1
1 -2 5 -13 34 -89 233 -610 1597 -4181
1 -3 11 -41 153 -571 2131 -7953 29681 -110771
1 -4 19 -91 436 -2089 10009 -47956 229771 -1100899
1 -5 29 -169 985 -5741 33461 -195025 1136689 -6625109
1 -6 41 -281 1926 -13201 90481 -620166 4250681 -29134601
1 -7 55 -433 3409 -26839 211303 -1663585 13097377 -103115431
1 -8 71 -631 5608 -49841 442961 -3936808 34988311 -310957991
1 -9 89 -881 8721 -86329 854569 -8459361 83739041 -828931049
MATHEMATICA
(* Array: *)
Grid[Table[LinearRecurrence[{-n, -1}, {1, 1 - n}, 10], {n, 10}]]
(*Array antidiagonals flattened (gives this sequence):*)
PROG
(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 */
CROSSREFS
Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269251, A269252, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
Cf. A094954 (unsigned version of this array, but missing the first row).
KEYWORD
sign,tabl
AUTHOR
STATUS
approved