A132191 Square array a(m,n) read by antidiagonals, defined by A000010(n)*a(m,n) = Sum_{k=1..n, gcd(k,n)=1} m^{ Sum_{d|n} A000010(d)/ (multiplicative order of k modulo d) }.

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 12, 18, 16, 5, 1, 12, 54, 40, 25, 6, 1, 40, 72, 160, 75, 36, 7, 1, 28, 405, 280, 375, 126, 49, 8, 1, 96, 390, 2176, 825, 756, 196, 64, 9, 1, 104, 1944, 2800, 8125, 2016, 1372, 288, 81, 10, 1, 280, 3411, 17920, 13175, 23976, 4312, 2304, 405

Offset

1,3

Comments

From Andrew Howroyd, Apr 22 2017: (Start)

Number of step shifted (decimated) sequences of length n using a maximum of m different symbols. See A056371 for an explanation of step shifts. -

Number of mappings with domain {0..n-1} and codomain {1..m} up to equivalence. Mappings A and B are equivalent if there is a d, prime to n, such that A(i) = B(i*d mod n) for i in {0..n-1}. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1830

Max Alekseyev, First 20 rows and columns of table

R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Example

Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 4, 6, 12, 12, 40, 28, 96, 104, 280, 216, 1248, 704, 2800, 4344, 8928, 8232, 44224, 29204, 136032, ...
3, 9, 18, 54, 72, 405, 390, 1944, 3411, 14985, 17802, 139968, 133104, 798525, 1804518, 5454378, 8072532, 64599849, 64573626, 437732424, ...
4, 16, 40, 160, 280, 2176, 2800, 17920, 44224, 263296, 419872, 4280320, 5594000, 44751616, 134391040, 539054080, 1073758360, 11453771776, 15271054960, 137575813120, ...
5, 25, 75, 375, 825, 8125, 13175, 103125, 327125, 2445625, 4884435, 61640625, 101732425, 1017323125, 3816215625, 19104609375, 47683838325, 635787765625, 1059638680675, 11924780390625, ...

Mathematica

a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n]==1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #]&], 0], {k, 1, n}]; Table[a[m-n+1, n], {m, 1, 15}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Dec 01 2015 *)

Prog

(PARI) for(i=1, 15, for(m=1, i, n=i-m+1; print1(sum(k=1, n, if(gcd(k, n)==1, m^sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d))), 0))/eulerphi(n)", "))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

Crossrefs

Row m=2 is A056371

Row m=3 is A056372

Row m=4 is A056373

Row m=5 is A056374

Row m=6 is A056375

Column n=2 is A000290

Column n=3 is A002411

Column n=4 is A019582

Cf. A285522, A285548.

Sequence in context: A142978 A152060 A093190 * A094437 A172431 A053123

Adjacent sequences: A132188 A132189 A132190 * A132192 A132193 A132194

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 01 2007, based on email from Max Alekseyev, Nov 08 2007

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

Offset corrected by Andrew Howroyd, Apr 20 2017

Status

approved