A065177
Table M(n,b) (columns: n >= 1, rows: b >= 0) gives the number of site swap juggling patterns with exact period n, using exactly b balls, where cyclic shifts are not counted as distinct.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 6, 3, 1, 0, 6, 15, 12, 4, 1, 0, 9, 42, 42, 20, 5, 1, 0, 18, 107, 156, 90, 30, 6, 1, 0, 30, 294, 554, 420, 165, 42, 7, 1, 0, 56, 780, 2028, 1910, 930, 273, 56, 8, 1, 0, 99, 2128, 7350, 8820, 5155, 1806, 420, 72, 9, 1, 0, 186, 5781, 26936
Offset: 0
Upper left corner starts as:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 3, 6, 9, 18, ...
1, 2, 6, 15, 42, 107, 294, ...
1, 3, 12, 42, 156, 554, 2028, ...
1, 4, 20, 90, 420, 1910, 8820, ...
1, 5, 30, 165, 930, 5155, 28830, ...
1, 6, 42, 273, 1806, 11809, 77658, ...
...
-
[seq(DistSS_table(j),j=0..119)]; DistSS_table := (n) -> DistSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2)));
with(numtheory); DistSS := proc(n,b) local d,s; s := 0; for d in divisors(n) do s := s+mobius(n/d)*((b+1)^d - b^d); od; RETURN(s/n); end;
-
trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
DistSS[n_, b_] := DivisorSum[n, MoebiusMu[n/#]*((b + 1)^# - b^#)&] /n;
a[n_] := DistSS[(((trinv[n] - 1)*(((1/2)*trinv[n]) + 1)) - n) + 1, (n - ((trinv[n]*(trinv[n] - 1))/2))];
Table[a[n], {n, 0, 119}] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)
A075147
Number of Lyndon words (aperiodic necklaces) with n beads of n colors.
Original entry on oeis.org
1, 1, 8, 60, 624, 7735, 117648, 2096640, 43046640, 999989991, 25937424600, 743008120140, 23298085122480, 793714765724595, 29192926025339776, 1152921504338411520, 48661191875666868480, 2185911559727674682148, 104127350297911241532840, 5242879999999487999992020
Offset: 1
-
with(numtheory):
a:= n-> add(n^d *mobius(n/d), d=divisors(n))/n:
seq(a(n), n=1..25); # Alois P. Heinz, Dec 21 2014
-
Table[Total@Map[ MoebiusMu[#1] n^(n/#1 - 1) &, Divisors[n]], {n, 20}] (* Olivier Gérard, Aug 05 2016 *)
-
a(n) = sumdiv(n, d, moebius(n/d) * n^d) / n; \\ Amiram Eldar, May 29 2025
A316073
a(n) is the n-th term of the inverse Weigh transform of j-> n^(j-1).
Original entry on oeis.org
1, 2, 6, 46, 420, 5185, 77658, 1376768, 28133616, 651325653, 16846515510, 481472773386, 15067838554860, 512473605894549, 18821719654854998, 742395982483047976, 31299550394528466960, 1404629090183809673484, 66851805805525048040334, 3363381327122907537090234
Offset: 1
-
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(g(i, k), j)*b(n-i*j, i-1,k), j=0..n/i)))
end:
g:= proc(n, k) option remember; k^(n-1)-b(n, n-1, k) end:
a:= n-> g(n$2):
seq(a(n), n=1..21);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
Sum[Binomial[g[i, k], j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = k^(n - 1) - b[n, n - 1, k];
a[n_] := g[n, n];
Array[a, 21] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)
A383042
Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Euler transform of j-> k^(j-1).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 15, 6, 0, 1, 5, 20, 42, 42, 9, 0, 1, 6, 30, 90, 156, 107, 18, 0, 1, 7, 42, 165, 420, 554, 294, 30, 0, 1, 8, 56, 273, 930, 1910, 2028, 780, 56, 0, 1, 9, 72, 420, 1806, 5155, 8820, 7350, 2128, 99, 0
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 6, 12, 20, 30, 42, ...
0, 3, 15, 42, 90, 165, 273, ...
0, 6, 42, 156, 420, 930, 1806, ...
0, 9, 107, 554, 1910, 5155, 11809, ...
0, 18, 294, 2028, 8820, 28830, 77658, ...
...
Showing 1-4 of 4 results.