A062160 Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.

0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, 1, 1, 0, 1, 0, 3, 2, 1, 0, -1, 1, 5, 7, 3, 1, 0, 1, 0, 11, 20, 13, 4, 1, 0, -1, 1, 21, 61, 51, 21, 5, 1, 0, 1, 0, 43, 182, 205, 104, 31, 6, 1, 0, -1, 1, 85, 547, 819, 521, 185, 43, 7, 1, 0, 1, 0, 171, 1640, 3277, 2604, 1111, 300, 57, 8, 1, 0, -1, 1, 341, 4921, 13107, 13021, 6665, 2101, 455, 73, 9, 1, 0

Offset

0,18

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

T(n, k) = n^(k-1)-n^(k-2)+n^(k-3)-...+(-1)^(k-1) = n^(k-1)-T(n, k-1) = nT(n, k-1)-(-1)^k = (n-1)T(n, k-1)+nT(n, k-2) = round[n^k/(n+1)] for n>1.

T(n,k) = (-1)^(k+1) * resultant( n*x + 1, (x^k-1)/(x-1) ). - Max Alekseyev, Sep 28 2021

G.f. of row n: x/((1+x) * (1-n*x)). - Seiichi Manyama, Apr 12 2019

E.g.f. of row n: (exp(n*x) - exp(-x))/(n+1). - Stefano Spezia, Feb 20 2024

Example

From Seiichi Manyama, Apr 12 2019: (Start)
Square array begins:
   0, 1, -1,  1,  -1,    1,    -1,      1, ...
   0, 1,  0,  1,   0,    1,     0,      1, ...
   0, 1,  1,  3,   5,   11,    21,     43, ...
   0, 1,  2,  7,  20,   61,   182,    547, ...
   0, 1,  3, 13,  51,  205,   819,   3277, ...
   0, 1,  4, 21, 104,  521,  2604,  13021, ...
   0, 1,  5, 31, 185, 1111,  6665,  39991, ...
   0, 1,  6, 43, 300, 2101, 14706, 102943, ... (End)

Mathematica

T[n_, k_]:=(n^k - (-1)^k)/(n+1); Join[{0}, Table[Reverse[Table[T[n-k, k], {k, 0, n}]], {n, 12}]]//Flatten (* Stefano Spezia, Feb 20 2024 *)

Crossrefs

Rows include A062157, A000035, A001045, A015518, A015521, A015531, A015540, A015552, A015565, A015577, A015585, A015592 etc.

Columns include A000004, A000012, A023443, A002061, A062158, A060884, A062159, A060888 etc.

Related to repunits in negative bases (cf. A055129 for positive bases).

Main diagonal gives A081216.

Sequence in context: A204065 A275281 A204176 * A301296 A194703 A343203

Adjacent sequences: A062157 A062158 A062159 * A062161 A062162 A062163

Keyword

sign,tabl

Author

Henry Bottomley, Jun 08 2001

Status

approved