A074930 -id:A074930 - cp's OEIS frontend

A338522 Number of cyclic Latin squares of order n.

1, 2, 12, 48, 480, 1440, 30240, 161280, 2177280, 14515200, 399168000, 1916006400, 74724249600, 523069747200, 10461394944000, 167382319104000, 5690998849536000, 38414242234368000, 2189611807358976000, 19463216065413120000, 613091306060513280000

Offset: 1

Eduard I. Vatutin, Nov 01 2020

A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places.

Equivalently, a Latin square is cyclic if and only if each row is a cyclic permutation of the first row and each column is a cyclic permutation of the first column.

			For n=5 there are 4 cyclic Latin squares with the first row in natural order:
  0 1 2 3 4   0 1 2 3 4   0 1 2 3 4   0 1 2 3 4
  1 2 3 4 0   2 3 4 0 1   3 4 0 1 2   4 0 1 2 3
  2 3 4 0 1   4 0 1 2 3   1 2 3 4 0   3 4 0 1 2
  3 4 0 1 2   1 2 3 4 0   4 0 1 2 3   2 3 4 0 1
  4 0 1 2 3   3 4 0 1 2   2 3 4 0 1   1 2 3 4 0
and 4*5! = 480 cyclic Latin squares.

Eduard I. Vatutin, Enumerating cyclic Latin squares and Euler totient function calculating using them, High-performance computing systems and technologies, 2020, Vol. 4, No. 2, pp. 40-48. (in Russian)
Eduard I. Vatutin, About the number of cyclic Latin squares and cyclic diagonal Latin squares (in Russian).

Cf. A000010, A000142, A074930, A123565.

a(n) = phi(n) * n!.

a(n) = A000010(n) * A000142(n).

A352620 Irregular triangle read by rows which are rows of successive n X n matrices M(n) with entries M(n)[i,j] = i*j mod n+1.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 1, 2, 3, 4, 2, 4, 1, 3, 3, 1, 4, 2, 4, 3, 2, 1, 1, 2, 3, 4, 5, 2, 4, 0, 2, 4, 3, 0, 3, 0, 3, 4, 2, 0, 4, 2, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 2, 4, 6, 1, 3, 5, 3, 6, 2, 5, 1, 4, 4, 1, 5, 2, 6, 3, 5, 3, 1, 6, 4, 2, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5

Offset: 1

Luca Onnis, Mar 24 2022

Each matrix represents all possible products between the elements of Z_(n+1), where Z_k is the ring of integers mod k.
Those matrices are symmetric.
The first row is equal to the first column which is equal to 1,2,...,n.

			Matrices begin:
  n=1:  1,
  n=2:  1, 2,
        2, 1,
  n=3:  1, 2, 3,
        2, 0, 2,
        3, 2, 1,
  n=4:  1, 2, 3, 4,
        2, 4, 1, 3,
        3, 1, 4, 2,
        4, 3, 2, 1;
For example, the 6 X 6 matrix generated by Z_7 is the following:
  1 2 3 4 5 6
  2 4 6 1 3 5
  3 6 2 5 1 4
  4 1 5 2 6 3
  5 3 1 6 4 2
  6 5 4 3 2 1
The trace of this matrix is 14 = A048153(7).

Onno Cain, Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
Matt Parker and Brady Haran, Finite Fields & Return of The Parker Square, Numberphile video (Oct 7, 2021).

Cf. A048153 (traces), A349099 (permanents), A160255 (sum entries), A088922 (ranks).

Cf. A074930.

Flatten[Table[Table[Mod[k*Table[i, {i, 1, p - 1}], p], {k, 1, p - 1}], {p, 1, 10}]]

A349099 a(n) is the permanent of the n X n matrix M(n) defined as M(n)[i,j] = i*j (mod n + 1).

Original entry on oeis.org

1, 1, 5, 32, 1074, 12600, 1525292, 34078720, 4072850100, 263459065600, 106809546673488, 2254519427530752, 3172225081523720416, 210351382651302645760, 45654014718074873700000, 11122845097194072534155264, 18156837198112938091803999360, 795289872611524024920215715840

Offset: 0

Stefano Spezia, Mar 25 2022

nonn

Det(M(n)) = 0 iff n = 4 or n > 5.
Rank(M(n)) = A088922(n+1).
Tr(M(n)) = A048153(n+1).

			See A352620 for the examples of matrix M(n).

Cf. A352620.

Cf. A048153, A074930, A088922, A160255.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
         Matrix(n, (i, j)-> (i*j) mod (n+1)))):
seq(a(n), n=0..16);  # Alois P. Heinz, Mar 25 2022

Join[{1},Table[Permanent[Table[Mod[j*Table[i, {i, n}], n+1], {j, n}]], {n, 17}]]

a(n) = matpermanent(matrix(n,n,i,j,(i*j)%(n+1))); \\ Michel Marcus, Mar 26 2022

A353192 Expansion of e.g.f. 1/(1 - Sum_{k>=1} phi(k) * x^k / k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 16, 110, 986, 10202, 126288, 1770120, 27939192, 489658632, 9455296896, 198951693360, 4537680805776, 111426422418768, 2931467216681856, 82273083792879744, 2453340521239749504, 77458777017799833216, 2581489882182061744128

Offset: 0

Seiichi Manyama, Apr 29 2022

nonn

Cf. A000010, A074930, A159929, A318917, A352887.

phi[k_] := phi[k] = EulerPhi[k]; a[0] = 1; a[n_] := a[n] = Sum[(k - 1)! * phi[k] * Binomial[n, k] * a[n - k], {k, 1, n}]; Array[a, 20, 0] (* Amiram Eldar, Apr 30 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, eulerphi(k)*x^k/k))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (j-1)!*eulerphi(j)*binomial(i, j)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A074930(k) * binomial(n,k) * a(n-k).

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A338522 Number of cyclic Latin squares of order n.

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A352620 Irregular triangle read by rows which are rows of successive n X n matrices M(n) with entries M(n)[i,j] = i*j mod n+1.

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A349099 a(n) is the permanent of the n X n matrix M(n) defined as M(n)[i,j] = i*j (mod n + 1).

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A353192 Expansion of e.g.f. 1/(1 - Sum_{k>=1} phi(k) * x^k / k), where phi is the Euler totient function A000010.

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