A179897 -id:A179897 - cp's OEIS frontend

A362783 Square array A(n,k) = (n^(2*k + 1) + 1)/(n + 1), n >= 0, k >= 0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 7, 1, 1, 1, 43, 61, 13, 1, 1, 1, 171, 547, 205, 21, 1, 1, 1, 683, 4921, 3277, 521, 31, 1, 1, 1, 2731, 44287, 52429, 13021, 1111, 43, 1, 1, 1, 10923, 398581, 838861, 325521, 39991, 2101, 57, 1, 1, 1, 43691, 3587227, 13421773, 8138021, 1439671

Offset: 0

Juri-Stepan Gerasimov, May 03 2023

			Array begins:
=====================================================================
n/k |  0    1      2       3         4          5            6   ...
----+----------------------------------------------------------------
0   |  1    1      1       1         1          1            1   ...
1   |  1    1      1       1         1          1            1   ...
2   |  1    3     11      43       171        683         2731   ...
3   |  1    7     61     547      4921      44287       398581   ...
4   |  1   13    205    3277     52429     838861     13421773   ...
5   |  1   21    521   13021    325521    8138021    203450521   ...
6   |  1   31   1111   39991   1439671   51828151   1865813431   ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=0..3 are A000012, A002061, A060884, A060888.
Rows n=2..4 are A007583, A066443, A299960.
Main diagonal is A179897.

/* as array */ [[&+[(-n)^j: j in [0..2*k]]: k in [0..6]]: n in [0..6]]; // Juri-Stepan Gerasimov, May 06 2023

A(n,k) = (n^(2*k + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023

A(n,k) = Sum_{j=0..2*k} (-n)^j.

a(49) corrected by Andrew Howroyd, Jan 20 2024

A280797 a(n) = (n^n - 1)*(n^n + 1)/(n + 1).

Original entry on oeis.org

0, 0, 5, 182, 13107, 1627604, 310968905, 84777884106, 31274997412295, 15009463529699912, 9090909090909090909, 6783562448903313426110, 6115142092568526471803195, 6552380728090615475599972572, 8231779712749862388415318790417, 11984441202055255416780710220336914

Offset: 0

Juri-Stepan Gerasimov, Jan 08 2017

			a(0) = 0 because (0^0 - 1)*(0^0 + 1)/(0 + 1) = 0,
a(1) = 0 because (1^1 - 1)*(1^1 + 1)/(1 + 1) = 0,
a(2) = 5 because (2^2 - 1)*(2^2 + 1)/(2 + 1) = 5.

Cf. A000312, A081216, A117812, A179897.

Magma
```
[(n^(2*n)-1)/(n+1): n in [0..15]];
```

Table[If[n == 0, 0, (n^n - 1) (n^n + 1)/(n + 1)], {n, 0, 15}] (* Michael De Vlieger, Jan 09 2017 *)

a(n) = A117812(n)/(n + 1).

Offset changed to 0 by Georg Fischer, Jul 15 2024

cp's OEIS Frontend

A362783 Square array A(n,k) = (n^(2*k + 1) + 1)/(n + 1), n >= 0, k >= 0, read by antidiagonals.

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