A133499 -id:A133499 - cp's OEIS frontend

A143324 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1).

1, 2, 0, 3, 2, 0, 4, 6, 6, 0, 5, 12, 24, 12, 0, 6, 20, 60, 72, 30, 0, 7, 30, 120, 240, 240, 54, 0, 8, 42, 210, 600, 1020, 696, 126, 0, 9, 56, 336, 1260, 3120, 4020, 2184, 240, 0, 10, 72, 504, 2352, 7770, 15480, 16380, 6480, 504, 0, 11, 90, 720, 4032, 16800, 46410, 78120, 65280, 19656, 990, 0

Offset: 1

Alois P. Heinz, Aug 07 2008

Column k is Dirichlet convolution of mu(n) with k^n.

The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k^2+k^4.

			T(2,3)=6, because there are 6 primitive words of length 2 over 3-letter alphabet {a,b,c}: ab, ac, ba, bc, ca, cb; note that the non-primitive words aa, bb and cc don't belong to the list; secondly note that the words in the list need not be Lyndon words, for example ba can be derived from ab by a cyclic rotation of the positions.
Table begins:
  1,  2,   3,    4,    5, ...
  0,  2,   6,   12,   20, ...
  0,  6,  24,   60,  120, ...
  0, 12,  72,  240,  600, ...
  0, 30, 240, 1020, 3120, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
C. J. Smyth, A coloring proof of a generalisation of Fermat's little theorem, Amer. Math. Monthly 93, No. 6 (1986), pp. 469-471.
Lucas Vieira Barbosa, Nonclassical Temporal Correlations Under Finite-Memory Constraints, Doctoral Thesis, Univ. Wien (Austria 2024). See p. 35.
Index entries for sequences related to Lyndon words

Columns k=1-10 give: A000007(n-1), A027375, A054718, A054719, A054720, A054721, A218124, A218125, A218126, A218127.
Rows n=1-10 give: A000027, A002378(k-1), A007531(k+1), A047928(k+1), A061167, A218130, A133499, A218131, A218132, A218133.
Main diagonal gives A252764.
Cf. A074650, A143325, A008683, A054525.

with(numtheory): f0:= proc(n) option remember; unapply(k^n-add(f0(d)(k), d=divisors(n)minus{n}), k) end; T:= (n,k)-> f0(n)(k); seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

f0[n_] := f0[n] = Function [k, k^n - Sum[f0[d][k], {d, Complement[Divisors[n], {n}]}]]; t[n_, k_] := f0[n][k]; Table[Table[t[n, 1 + d - n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

T(n,k) = Sum_{d|n} k^d * mu(n/d).

T(n,k) = k^n - Sum_{d

T(n,k) = A143325(n,k) * k.

T(n,k) = A074650(n,k) * n.

So Sum_{d|n} k^d * mu(n/d) == 0 (mod n), this is a generalization of Fermat's little theorem k^p - k == 0 (mod p) for primes p to an arbitrary modulus n (see the Smyth link). - Franz Vrabec, Feb 09 2021

A106512 Array read by antidiagonals: a(n,k) = number of k-colorings of a circle of n nodes (n >= 1, k >= 1).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 12, 6, 2, 0, 0, 20, 24, 18, 0, 0, 0, 30, 60, 84, 30, 2, 0, 0, 42, 120, 260, 240, 66, 0, 0, 0, 56, 210, 630, 1020, 732, 126, 2, 0, 0, 72, 336, 1302, 3120, 4100, 2184, 258, 0, 0, 0, 90, 504, 2408, 7770, 15630, 16380, 6564, 510, 2, 0, 0, 110

Offset: 1

Joshua Zucker, May 29 2005

Note that we keep one edge in the circular graph even when there's only one node (so there are 0 colorings of one node with k colors).

Number of closed walks of length n on the complete graph K_{k}. - Andrew Howroyd, Mar 12 2017

			From _Andrew Howroyd_, Mar 12 2017: (Start)
Table begins:
  0 0   0     0      0       0        0        0         0 ...
  0 2   6    12     20      30       42       56        72 ...
  0 0   6    24     60     120      210      336       504 ...
  0 2  18    84    260     630     1302     2408      4104 ...
  0 0  30   240   1020    3120     7770    16800     32760 ...
  0 2  66   732   4100   15630    46662   117656    262152 ...
  0 0 126  2184  16380   78120   279930   823536   2097144 ...
  0 2 258  6564  65540  390630  1679622  5764808  16777224 ...
  0 0 510 19680 262140 1953120 10077690 40353600 134217720 ...
(End)
a(4,3) = 18 because there are three choices for the first node's color (call it 1) and then two choices for the second node's color (call it 2) and then the remaining two nodes can be 12, 13, or 32. So in total there are 3*2*3 = 18 ways. a(3,4) = 4*3*2 = 24 because the three nodes must be three distinct colors.

Andrew Howroyd, Table of n, a(n) for n = 1..1274

Columns include A092297, A226493. Main diagonal is A118537.
Rows 2-7 are A002378, A007531, A091940, A061167, A131472, A133499.
Cf. A090860, A208535.

a(n, k) = (k-1)^n + (-1)^n * (k-1).

a(67) corrected by Andrew Howroyd, Mar 12 2017

A030180 a(n) = (n^7 - n)/42.

Original entry on oeis.org

0, 0, 3, 52, 390, 1860, 6665, 19608, 49932, 113880, 238095, 463980, 853138, 1494012, 2509845, 4068080, 6391320, 9769968, 14576667, 21282660, 30476190, 42883060, 59389473, 81067272, 109201700, 145321800, 191233575, 249056028, 321260202, 410711340, 520714285

Offset: 0

Charlton Harrison (charlton(AT)bach.dynet.com)

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A133499 (n^7-n).

List([0..35], n-> (n^7-n)/42); # G. C. Greubel, Dec 28 2019

[(n^7-n)/42: n in [0..35]]; // G. C. Greubel, Dec 28 2019

seq( (n^7-n)/42, n=0..35); # G. C. Greubel, Dec 28 2019

Table[(n^7-n)/42, {n,0,35}] (* G. C. Greubel, Dec 28 2019 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,0,3,52,390,1860,6665,19608},40] (* Harvey P. Dale, May 14 2022 *)

a(n) = (n^7-n)/42; \\ Michel Marcus, Aug 30 2013

[(n^7-n)/42 for n in (0..35)] # G. C. Greubel, Dec 28 2019

a(n) = A133499(n)/42. - Michel Marcus, Aug 30 2013
From Stefano Spezia, Dec 29 2019: (Start)
O.g.f.: x^2*(3 + 28*x + 58*x^2 + 28*x^3 + 3*x^4)/(1 - x)^8.
E.g.f.: (1/42)*exp(x)*x^2*(63 + 301*x + 350*x^2 + 140*x^3 + 21*x^4 + x^5).
(End)

A363916 Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 6, 6, 4, 1, 0, 0, 12, 24, 12, 5, 1, 0, 0, 30, 72, 60, 20, 6, 1, 0, 0, 54, 240, 240, 120, 30, 7, 1, 0, 0, 126, 696, 1020, 600, 210, 42, 8, 1, 0, 0, 240, 2184, 4020, 3120, 1260, 336, 56, 9, 1

Offset: 0

Peter Luschny, Jul 04 2023

Row n gives the number of n-ary sequences with primitive period k.

See A074650 and A143324 for combinatorial interpretations.

			Array A(n, k) starts:
[0] 1, 0,  0,   0,    0,     0,      0,       0,        0, ... A000007
[1] 1, 1,  0,   0,    0,     0,      0,       0,        0, ... A019590
[2] 1, 2,  2,   6,   12,    30,     54,     126,      240, ... A027375
[3] 1, 3,  6,  24,   72,   240,    696,    2184,     6480, ... A054718
[4] 1, 4, 12,  60,  240,  1020,   4020,   16380,    65280, ... A054719
[5] 1, 5, 20, 120,  600,  3120,  15480,   78120,   390000, ... A054720
[6] 1, 6, 30, 210, 1260,  7770,  46410,  279930,  1678320, ... A054721
[7] 1, 7, 42, 336, 2352, 16800, 117264,  823536,  5762400, ... A218124
[8] 1, 8, 56, 504, 4032, 32760, 261576, 2097144, 16773120, ... A218125
A000012|A002378| A047928   |   A218130     |      A218131
    A001477,A007531,    A061167,        A133499,   (diagonal A252764)
.
Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 0,  2,   1;
[4] 0, 0,  2,   3,   1;
[5] 0, 0,  6,   6,   4,   1;
[6] 0, 0, 12,  24,  12,   5,  1;
[7] 0, 0, 30,  72,  60,  20,  6, 1;
[8] 0, 0, 54, 240, 240, 120, 30, 7, 1;

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Variant: A143324.
Rows: A000007 (n=0), A019590 (n=1), A027375 (n=2), A054718 (n=3), A054719 (n=4), A054720, A054721, A218124, A218125.
Columns: A000012 (k=0), A001477 (k=1), A002378 (k=2), A007531(k=3), A047928, A061167, A218130, A133499, A218131.
Cf. A252764 (main diagonal), A074650, A363914.

A363916 := (n, k) -> local d; add(A363914(k, d) * n^d, d = 0 ..k):
for n from 0 to 9 do seq(A363916(n, k), k = 0..8) od;

def A363916(n, k): return sum(A363914(k, d) * n^d for d in range(k + 1))
for n in range(9): print([A363916(n, k) for k in srange(9)])
def T(n, k): return A363916(k, n - k)

If k > 0 then k divides A(n, k), see the transposed array of A074650.

If k > 0 then n divides A(n, k), see the transposed array of A143325.

Showing 1-4 of 4 results.

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A143324 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1).

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A106512 Array read by antidiagonals: a(n,k) = number of k-colorings of a circle of n nodes (n >= 1, k >= 1).

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A030180 a(n) = (n^7 - n)/42.

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A363916 Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

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