A158388 -id:A158388 - cp's OEIS frontend

A165795 Array A(n, k) = numerator of 1/n^2 - 1/k^2 with A(0,k) = 1 and A(n,0) = -1, read by antidiagonals.

1, -1, 1, -1, 0, 1, -1, -3, 3, 1, -1, -8, 0, 8, 1, -1, -15, -5, 5, 15, 1, -1, -24, -3, 0, 3, 24, 1, -1, -35, -21, -7, 7, 21, 35, 1, -1, -48, -2, -16, 0, 16, 2, 48, 1, -1, -63, -45, -1, -9, 9, 1, 45, 63, 1, -1, -80, -15, -40, -5, 0, 5, 40, 15, 80, 1, -1, -99, -77, -55, -33, -11, 11, 33, 55, 77, 99, 1

Offset: 0

Paul Curtz, Sep 27 2009

A row of A(0,k)= 1 is added on top of the array shown in A172157, which is then read upwards by antidiagonals.

One may also interpret this as appending a 1 to each row of A173651 or adding a column of -1's and a diagonal of +1's to A165507.

			The array, A(n, k), of numerators starts in row n=0 with columns m>=0 as:
  .1...1...1...1...1...1...1...1...1...1...1.
  -1...0...3...8..15..24..35..48..63..80..99. A005563, A147998
  -1..-3...0...5...3..21...2..45..15..77...6. A061037, A070262
  -1..-8..-5...0...7..16...1..40..55...8..91. A061039
Antidiagonal triangle, T(n, k), begins as:
   1;
  -1,   1;
  -1,   0,   1;
  -1,  -3,   3,  1;
  -1,  -8,   0,  8,  1;
  -1, -15,  -5,  5, 15,  1;
  -1, -24,  -3,  0,  3, 24,  1;
  -1, -35, -21, -7,  7, 21, 35, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A158388, A165507, A172157, A173651.

Cf. A005408, A005563, A061037, A061039, A070262, A147998.

T[n_, k_]:= If[k==n, 1, If[k==0, -1, Numerator[1/(n-k)^2 - 1/k^2]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 10 2022 *)

def A165795(n,k):
    if (k==n): return 1
    elif (k==0): return -1
    else: return numerator(1/(n-k)^2 -1/k^2)
flatten([[A165795(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 10 2022

A(n, k) = numerator(1/n^2 - 1/k^2) with A(0,k) = 1 and A(n,0) = -1 (array).
A(n, 0) = -A158388(n).
A(n, k) = A172157(n,k), n>=1.
From G. C. Greubel, Mar 10 2022: (Start)
T(n, k) = numerator(1/(n-k)^2 -1/k^2), with T(n,n) = 1, T(n,0) = -1 (triangle).
A(n, n) = T(2*n, n) = 0^n.
Sum_{k=0..n} T(n, k) = 0^n.
T(n, n-k) = -T(n,k).
T(2*n+1, n) = -A005408(n). (End)

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

cp's OEIS Frontend

A165795 Array A(n, k) = numerator of 1/n^2 - 1/k^2 with A(0,k) = 1 and A(n,0) = -1, read by antidiagonals.

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