A294099 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.

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, 8, 41, 139, 265, 199, 13, 2, 1, 9, 55, 239, 666, 989, 521, 15, 1, 1, 10, 71, 377, 1393, 3191, 3691, 1364, 17, -1, 1, 11, 89, 559, 2584, 8119, 15289, 13775, 3571, 19, -2

Offset

1,3

Comments

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.

For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

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

Table of n, a(n) for n=1..66.

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Formula

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.

G.f. for row n: (1 + x)/(1 - n*x + x^2), n >= 1.

A(n, k) = B(-n, k) where B = A29045. - Michael Somos, Jun 19 2023

Example

Array begins:
1   2    1    -1     -2      -1        1         2          1          -1
1   3    5     7      9      11       13        15         17          19
1   4   11    29     76     199      521      1364       3571        9349
1   5   19    71    265     989     3691     13775      51409      191861
1   6   29   139    666    3191    15289     73254     350981     1681651
1   7   41   239   1393    8119    47321    275807    1607521     9369319
1   8   55   377   2584   17711   121393    832040    5702887    39088169
1   9   71   559   4401   34649   272791   2147679   16908641   133121449
1  10   89   791   7030   62479   555281   4935050   43860169   389806471
1  11  109  1079  10681  105731  1046629  10360559  102558961  1015229051

Mathematica

(* Array: *)
Grid[Table[LinearRecurrence[{n, -1}, {1, 1 + n}, 10], {n, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
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}]]

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, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.

Rows: A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, ...

Columns: A000012, A000027, A028387, ...

Sequence in context: A133913 A209485 A209344 * A209115 A353430 A353391

Adjacent sequences: A294096 A294097 A294098 * A294100 A294101 A294102

Keyword

sign,tabl

Author

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Oct 22 2017

Status

approved