A065177 -id:A065177 - cp's OEIS frontend

A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

1, 2, 2, 3, 4, 4, 4, 1, 6, 6, 5, 6, 2, 8, 8, 6, 3, 8, 4, 10, 10, 7, 8, 1, 10, 6, 12, 12, 8, 5, 10, 2, 12, 8, 14, 14, 9, 10, 3, 12, 4, 14, 10, 16, 16, 10, 7, 12, 1, 14, 6, 16, 12, 18, 18, 11, 12, 5, 14, 2, 16, 8, 18, 14, 20, 20, 12, 9, 14, 3, 16, 4, 18, 10, 20, 16, 22, 22, 13, 14, 7, 16, 1

Offset: 0

Antti Karttunen, Oct 19 2001

Simple periodic site swap permutations of natural numbers.

Row n of the table (starting from n=0) gives a permutation of natural numbers corresponding to the simple, infinite, periodic site swap pattern ...nnnnn...

			Table begins:
1 2 3 4 5 6 7 ...
2 4 1 6 3 8 5 ...
4 6 2 8 1 10 3 ...
6 8 4 10 2 12 1 ...

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

Successive rows and associated site swap sequences, starting from the zeroth row: (A000027, A000004), (A065164, A000012), (A065165, A007395), (A065166, A010701). Cf. also A065171, A065174, A065177. trinv given at A054425.

PerSS_table := (n) -> PerSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2))); PerSS := (n,c) -> Z2N(N2Z(n)+c);
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
[seq(PerSS_table(j),j=0..119)];

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then the n-th term of the k-th row is f(g(n)+k).

A065178 Number of site swap patterns with 2 balls and exact period n.

Original entry on oeis.org

1, 2, 6, 15, 42, 107, 294, 780, 2128, 5781, 15918, 43885, 122010, 340323, 954394, 2685930, 7588770, 21507696, 61144062, 174283887, 498012094, 1426213191, 4092816966, 11767176070, 33890202192, 97761428205, 282424564744

Offset: 1

Antti Karttunen, Oct 19 2001

nonn

When interspersed with 0's, exponents in expansion of A065481 as a product zeta(n)^(-a(n)).

			We have one period 1 (2), two period 2 (31/13 and 40/04) and six period three 2-ball siteswaps (312, 330, 411, 420, 501, 600) (The average of the digits is always 2).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Juggling Information Service, Site Swap FAQs
M. Macauley, Braids and Juggling patterns, Thesis (Harvey Mudd College) 2003, eq. (A1).
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Row 2 of A065177.

Cf. A008683, A065179, A065180, A065481.

[seq(DistSS(p,2),p=1..60)];
A065178 := proc(n)
    add( mobius(n/d)*(3^d-2^d),d=numtheory[divisors](n)) /n ;
end proc:
seq(A065178(n),n=1..30) ; # R. J. Mathar, Aug 05 2015

a[n_] := DivisorSum[n, MoebiusMu[n/#] * (3^#-2^#)&] / n; Array[a, 30] (* Jean-François Alcover, Mar 05 2016, after R. J. Mathar *)

a(n) ~ 3^n/n. - Vaclav Kotesovec, Mar 05 2016
Inverse Euler transform of A133494. - Alois P. Heinz, Jun 23 2018
G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 3*x^k))/k. - Seiichi Manyama, Apr 14 2025

A065180 Number of site swap patterns with 4 balls and exact period n.

Original entry on oeis.org

1, 4, 20, 90, 420, 1910, 8820, 40590, 187880, 871494, 4057620, 18945960, 88738020, 416787030, 1962922276, 9268287390, 43868210820, 208109782580, 989400443220, 4713395564772, 22497100553820, 107572434560790, 515241748300020

Offset: 1

Antti Karttunen, Oct 19 2001

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Juggling Information Service, Site Swap FAQs

Row 4 of A065177.

Cf. A008683, A065178, A065179.

[seq(DistSS(p,4),p=1..60)];
A065180 := proc(n)
    add( mobius(n/d)*(5^d-4^d),d=numtheory[divisors](n)) /n ;
end proc:
seq(A065180(n),n=1..30) ; # R. J. Mathar, Aug 05 2015

a[n_] := DivisorSum[n, MoebiusMu[n/#]*(5^# - 4^#)&]/n; Array[a, 25] (* Jean-François Alcover, Mar 06 2016 *)

G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 5*x^k))/k. - Seiichi Manyama, Apr 14 2025

A065179 Number of site swap patterns with 3 balls and exact period n.

Original entry on oeis.org

1, 3, 12, 42, 156, 554, 2028, 7350, 26936, 98874, 365196, 1353520, 5039580, 18831306, 70626140, 265741350, 1002984060, 3796211692, 14406086604, 54801192684, 208932673508, 798218225802, 3055417434732, 11716355452900

Offset: 1

Antti Karttunen, Oct 19 2001

nonn

			We have one period 1 (3, the three-ball cascade), three period two (42/24, 51/15 = three-ball shower and 60/06) and twelve period three 3-ball siteswaps (423, 441, 450, 522, 531, 603, 612, 630, 711, 720, 801, 900) (The average of digits is always 3).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Juggling Information Service, Site Swap FAQs
M. Macauley, Braids and Juggling patterns, Thesis (Harvey Mudd College) 2003, equation A1.

Row 3 of A065177.

Cf. A008683, A065178, A065180.

[seq(DistSS(p,3),p=1..60)];
A065179 := proc(n)
    add( mobius(n/d)*(4^d-3^d),d=numtheory[divisors](n)) /n ;
end proc:
seq(A065179(n),n=1..30) ; # R. J. Mathar, Aug 05 2015

a[n_] := DivisorSum[n, MoebiusMu[n/#]*(4^# - 3^#)&]/n; Array[a, 25] (* Jean-François Alcover, Mar 06 2016 *)

G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 4*x^k))/k. - Seiichi Manyama, Apr 14 2025

A306173 a(n) is the n-th term of the inverse Euler transform of j-> n^(j-1).

Original entry on oeis.org

1, 1, 6, 42, 420, 5155, 77658, 1376340, 28133616, 651317463, 16846515510, 481472570920, 15067838554860, 512473599799551, 18821719654854998, 742395982266536520, 31299550394528466960, 1404629090174946183156, 66851805805525048040334, 3363381327122496643643628

Offset: 1

Alois P. Heinz, Jun 23 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..200

Main diagonal of A065177.

Cf. A075147, A316073.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, k)+j-1, 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 - 1, 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];
a /@ Range[21] (* Jean-François Alcover, Jan 06 2020, after Alois P. Heinz *)

a(n) ~ (1 - exp(-1)) * n^(n-1). - Vaclav Kotesovec, Oct 08 2019

A071160 Łukasiewicz words that are also valid asynchronic siteswap juggling patterns.

Original entry on oeis.org

0, 1, 20, 11, 300, 201, 120, 111, 4000, 3001, 2020, 2011, 1300, 1201, 1120, 1111, 50000, 40001, 30020, 30011, 20300, 20201, 20120, 20111, 14000, 13001, 12020, 12011, 11300, 11201, 11120, 11111, 600000, 500001, 400020, 400011, 300300

Offset: 0

Antti Karttunen, May 14 2002

Note: this finite decimal representation works only up to the 511th term, as the 512th such word is already (10,0,0,0,0,0,0,0,0,0). The sequence A071161 shows the initial portion of this sequence sorted.

Peter J. Beek and Arthur Lewbel, The Science of Juggling, Scientific American, Nov, 1995, Vol. 273, Number 5, pp. 92-97.
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
A. Karttunen, Gatomorphisms and other excursions amidst the plane trees and parenthesizations (Includes the complete Scheme program for computing this sequence)
R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz

Subset of A071153.

Cf. A060495, A060498, A065177, A071162, A071163.

Construction: starting from the most significant (the leftmost) bit, replace each 1-bit in the binary expansion of n with the distance to the next 1-bit to the right, allowing a cyclic wrap-over from the least-significant 1-bit to the most significant 1-bit. I.e. from 22 = 10110 in binary we get 20120, the 22nd term of this sequence.

a(n) = A071161(A054429(n)).

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

Seiichi Manyama, Apr 13 2025

			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, ...
  ...

Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Columns k=1..5 give A000007, A059966, A065178, A065179, A065180.
Main diagonal gives A306173.
Cf. A065177 (another version).
Cf. A008683, A074650, A383033.

a(n, k) = sumdiv(n, d, moebius(n/d)*(k^d-(k-1)^d))/n;

A(n,k) = (1/n) * Sum_{d|n} mu(n/d) * (k^d - (k-1)^d).

A(n,k) = (1/n) * (k^n - (k-1)^n - Sum_{d

A(n,k) = A074650(n,k) - A074650(n,k-1).

Product_{n>=1} 1/(1 - x^n)^A(n,k) = (1 - (k-1)*x)/(1 - k*x).

G.f. of column k: Sum_{j>=1} mu(j) * log(1 + x^j/(1 - k*x^j)) / j.

Showing 1-7 of 7 results.

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A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

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A065178 Number of site swap patterns with 2 balls and exact period n.

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A065180 Number of site swap patterns with 4 balls and exact period n.

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A065179 Number of site swap patterns with 3 balls and exact period n.

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A306173 a(n) is the n-th term of the inverse Euler transform of j-> n^(j-1).

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A071160 Łukasiewicz words that are also valid asynchronic siteswap juggling patterns.

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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).

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