cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Antti Karttunen, Oct 19 2001

Keywords

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

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.

Programs

  • 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 *)

Formula

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

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

Views

Author

Antti Karttunen, Oct 19 2001

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

Row 2 of A065177.
Cf. A008683, A065179, A065180, A065481.

Programs

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

Formula

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

Views

Author

Antti Karttunen, Oct 19 2001

Keywords

Links

Crossrefs

Row 4 of A065177.
Cf. A008683, A065178, A065179.

Programs

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

Formula

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

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

Views

Author

Seiichi Manyama, Apr 13 2025

Keywords

Examples

			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, ...
  ...

Links

Crossrefs

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

Programs

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

Formula

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 1-4 of 4 results.

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