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

0,8

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

Formula

Row n is the inverse Euler transform of j-> n^(j-1). - Alois P. Heinz, Jun 23 2018

Example

Upper left corner starts as:
1, 0,  0,  0,   0, ...
1, 1,  2,  3,   6, ...
1, 2,  6, 15,  42, ...
1, 3, 12, 42, 156, ...
1, 4, 20, 90, 420, ...
...

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;

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 *)

Crossrefs

Row 1: A059966, row 2: A065178, row 3: A065179, row 4: A065180.

Column 1: A002378, column 2: A059270.

Main diagonal gives A306173.

Cf. also A065167. trinv given at A054425.

Sequence in context: A155161 A185937 A292086 * A064044 A213980 A349373

Adjacent sequences: A065174 A065175 A065176 * A065178 A065179 A065180

Keyword

nonn,tabl

Author

Antti Karttunen, Oct 19 2001

Status

approved