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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.
15
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, -1, 1, 1, -6, 29, -91, 153, -89, 1, 0, 1, -7, 41, -169, 436, -571, 233, -1, -1, 1, -8, 55, -281, 985, -2089, 2131, -610, 1, 1, 1, -9, 71, -433, 1926, -5741, 10009, -7953, 1597, -1, 0
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
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):*)
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}]]
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. A094954 (unsigned version of this array, but missing the first row).
Sequence in context: A140075 A099555 A262082 * A124530 A243631 A070914
KEYWORD
sign,tabl
AUTHOR
STATUS
approved