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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.

%I A065177 #22 Apr 14 2025 07:40:16
%S A065177 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,
%T A065177 107,156,90,30,6,1,0,30,294,554,420,165,42,7,1,0,56,780,2028,1910,930,
%U A065177 273,56,8,1,0,99,2128,7350,8820,5155,1806,420,72,9,1,0,186,5781,26936
%N 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.
%H A065177 G. C. Greubel, <a href="/A065177/b065177.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A065177 Joe Buhler and R. L. Graham, <a href="http://www.cecm.sfu.ca/organics/papers/buhler/index.html">Juggling Drops and Descents</a>, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
%H A065177 Juggling Information Service, <a href="http://www.juggling.org/bin/mfs/JIS/help/siteswap/">Site Swap FAQs</a>
%F A065177 Row n is the inverse Euler transform of j-> n^(j-1). - _Alois P. Heinz_, Jun 23 2018
%e A065177 Upper left corner starts as:
%e A065177   1, 0,  0,   0,    0,     0,     0, ...
%e A065177   1, 1,  2,   3,    6,     9,    18, ...
%e A065177   1, 2,  6,  15,   42,   107,   294, ...
%e A065177   1, 3, 12,  42,  156,   554,  2028, ...
%e A065177   1, 4, 20,  90,  420,  1910,  8820, ...
%e A065177   1, 5, 30, 165,  930,  5155, 28830, ...
%e A065177   1, 6, 42, 273, 1806, 11809, 77658, ...
%e A065177   ...
%p A065177 [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)));
%p A065177 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;
%t A065177 trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
%t A065177 DistSS[n_, b_] := DivisorSum[n, MoebiusMu[n/#]*((b + 1)^# - b^#)&] /n;
%t A065177 a[n_] := DistSS[(((trinv[n] - 1)*(((1/2)*trinv[n]) + 1)) - n) + 1, (n - ((trinv[n]*(trinv[n] - 1))/2))];
%t A065177 Table[a[n], {n, 0, 119}] (* _Jean-François Alcover_, Mar 06 2016, adapted from Maple *)
%Y A065177 Row 1: A059966, row 2: A065178, row 3: A065179, row 4: A065180.
%Y A065177 Column 1: A002378, column 2: A059270.
%Y A065177 Main diagonal gives A306173.
%Y A065177 Cf. also A065167. trinv given at A054425.
%K A065177 nonn,tabl
%O A065177 0,8
%A A065177 _Antti Karttunen_, Oct 19 2001