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.
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
Examples
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, ... ...
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
- Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
- Juggling Information Service, Site Swap FAQs
Crossrefs
Programs
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Maple
[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;
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Mathematica
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 *)
Formula
Row n is the inverse Euler transform of j-> n^(j-1). - Alois P. Heinz, Jun 23 2018