A320095 -id:A320095 - cp's OEIS frontend

A143327 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words (n,k >= 1) with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 11, 1, 1, 5, 19, 35, 26, 1, 1, 6, 29, 79, 115, 53, 1, 1, 7, 41, 149, 334, 347, 116, 1, 1, 8, 55, 251, 773, 1339, 1075, 236, 1, 1, 9, 71, 391, 1546, 3869, 5434, 3235, 488, 1, 1, 10, 89, 575, 2791, 9281, 19493, 21754, 9787, 983, 1, 1, 11

Offset: 1

Alois P. Heinz, Aug 07 2008

The coefficients of the polynomial of row n are given by the n-th row of triangle A134541; for example row 4 has polynomial -1+k^2+k^3.

			T(3,3) = 11, because 11 words of length <=3 over 3-letter alphabet {a,b,c} are primitive and earlier than others derived by cyclic shifts of the alphabet: a, ab, ac, aab, aac, aba, abb, abc, aca, acb, acc.
Table begins:
  1,   1,    1,     1,     1,      1,      1,       1, ...
  1,   2,    3,     4,     5,      6,      7,       8, ...
  1,   5,   11,    19,    29,     41,     55,      71, ...
  1,  11,   35,    79,   149,    251,    391,     575, ...
  1,  26,  115,   334,   773,   1546,   2791,    4670, ...
  1,  53,  347,  1339,  3869,   9281,  19543,   37367, ...
  1, 116, 1075,  5434, 19493,  55936, 137191,  299510, ...
  1, 236, 3235, 21754, 97493, 335656, 960391, 2396150, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Lyndon words

Columns k=1-10 give: A000012, A085945, A320087, A320088, A320089, A320090, A320091, A320092, A320093, A320094.
Rows n=1-4 give: A000012, A000027, A028387, A003777.
Main diagonal gives A320095.
Cf. A143325, A143326, A134541, A008683.

with(numtheory):
f1:= proc (n) option remember; unapply(k^(n-1)
        -add(f1(d)(k), d=divisors(n) minus {n}), k)
     end:
g1:= proc(n) option remember; unapply(add(f1(j)(x), j=1..n), x) end:
T:= (n, k)-> g1(n)(k):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

t[n_, k_] := Sum[k^(d-1)*MoebiusMu[j/d], {j, 1, n}, {d, Divisors[j]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 13 2013 *)

T(n,k) = Sum_{j=1..n} Sum_{d|j} k^(d-1) * mu(j/d).
T(n,k) = Sum_{j=1..n} A143325(j,k).
T(n,k) = A143326(n,k) / k.

A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

Original entry on oeis.org

1, 4, 15, 92, 790, 9384, 137326, 2397352, 48428487, 1111122360, 28531183329, 810554859732, 25239592620853, 854769763924104, 31278135039463245, 1229782938533709200, 51702516368332126932, 2314494592676172411516, 109912203092257573556274, 5518821052632117898282620

Offset: 1

Ilya Gutkovskiy, Feb 16 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A024916, A268235, A308313, A308814, A319194, A320095.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), this sequence (q=n).

A332533:= func< n | (&+[Floor(n/j)*n^(j-1): j in [1..n]]) >;
[A332533(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024

seq(add(n^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023

Table[(1/n) Sum[Floor[n/k] n^k, {k, 1, n}], {n, 1, 20}]
Table[(1/n) Sum[Sum[n^d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[x^k/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]

a(n) = sum(k=1, n, (n\k)*n^k)/n; \\ Michel Marcus, Feb 16 2020

a(n) = sum(k=1, n, sumdiv(k, d, n^(d-1))); \\ Seiichi Manyama, May 29 2021

def A332533(n): return sum((n//j)*n^(j-1) for j in range(1,n+1))
[A332533(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k / (1 - n*x^k).
a(n) = (1/n) * Sum_{k=1..n} Sum_{d|k} n^d.
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 28 2021
a(n) = (1/(n-1)) * Sum_{k=1..n} (n^floor(n/k) - 1), for n>=2. - Ridouane Oudra, Mar 05 2023

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A143327 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words (n,k >= 1) with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

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