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 1-3 of 3 results.

A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 0, -3, -6, 5, 61, 126, -308, -2772, -5669, 25630, 224730, 486551, -3068155, -29264219, -72173176, 513535711, 5625869262, 16687752839, -113740116822, -1496118902963, -5508392724427, 31534346503605, 523333047780288, 2414704077547660, -10254467367668159
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 11 2018

Keywords

Comments

Exponential transform of A008683.

Examples

			E.g.f.: A(x) = 1 + x/1! - 3*x^3/3! - 6*x^4/4! + 5*x^5/5! + 61*x^6/6! + 126*x^7/7! - 308*x^8/8! - 2772*x^9/9! - 5669*x^10/10! + ...

Links

Crossrefs

Cf. A008683, A050385, A050386, A068341, A073776, A104688, A117209, A117210, A195589, A204290, A300663.

Programs

  • Mathematica
    nmax = 26; CoefficientList[Series[Exp[Sum[MoebiusMu[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[MoebiusMu[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022

Formula

E.g.f.: exp(Sum_{k>=1} A008683(k)*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022

A124839 Inverse binomial transform of the Moebius sequence {mu(k), k >= 1}, A008683.

Original entry on oeis.org

1, -2, 2, -1, -2, 10, -30, 76, -173, 363, -717, 1363, -2551, 4797, -9189, 18015, -36008, 72725, -146930, 294423, -581758, 1130231, -2158552, 4061201, -7557522, 13983585, -25872679, 48115364, -90273986, 171186911, -328120527, 635014942, -1239093092, 2434924044
Offset: 1

Views

Author

Gary W. Adamson, Nov 10 2006

Keywords

Comments

Left border of finite difference table of Moebius sequence A008683.
This is also the inverse binomial transform of (0, {A002321(n), n=1,2,...}), where A002321(n) is Mertens's function. - Tilman Neumann, Dec 13 2008

Examples

			Given (1, -1, -1, 0, -1, ...), taking finite differences, we obtain the array whose left border is the present sequence.
       1,   -1,   -1,    0,   -1,    1,   -1, ...
         -2,    0,    1,   -1,    2,   -2, ...
             2,    1,   -2,    3,   -4, ...
               -1,   -3,    5,   -7, ...
                  -2,    8,  -12, ...
                     10,  -20, ...
                       -30, ...

Crossrefs

Cf. A002321, A008683, A104688, A124840.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n-k) * Binomial[n-1, k-1] * MoebiusMu[k], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Jun 01 2025 *)

Formula

For n >= 1, a(n) = Sum_{k=0..n-1} (-1)^(n-1-k)*binomial(n-1,k)*mu(k+1). - N. J. A. Sloane, Nov 23 2022

Extensions

More terms from Tilman Neumann, Dec 13 2008
Edited by N. J. A. Sloane, Nov 23 2022
More terms from Amiram Eldar, Jun 01 2025

A127514 Binomial transform of an infinite lower triangular matrix with mu(n) in the diagonal.

Original entry on oeis.org

1, 1, -1, 1, -2, -1, 1, -3, -3, 0, 1, -4, -6, 0, -1, 1, -5, -10, 0, -5, 1, 1, -6, -15, 0, -15, 6, -1, 1, -7, -21, 0, -35, 21, -7, 0, 1, -8, -28, 0, -70, 56, -28, 0, 0, 1, -9, -36, 0, -126, 126, -84, 0, 0, 1, 1, -10, -45, 0, -210, 252, -210, 0, 0, 10, -1
Offset: 1

Views

Author

Gary W. Adamson, Jan 17 2007

Keywords

Comments

Right border = mu(n).
Row sums = A104688, the binomial transform of mu(n): 1, 0, -2, -5, -10, -18, ...

Examples

			First few rows of the triangle:
  1;
  1, -1;
  1, -2,  -1;
  1, -3,  -3, 0;
  1, -4,  -6, 0, -1;
  1, -5, -10, 0, -5, 1;
  ...

Crossrefs

Cf. A007318, A008683, A104688.
Cf. A127512 (M*P).

Programs

  • PARI
    row(n) = {my(M = matrix(n, n, i, j, if (i==j, moebius(i))), P = matrix(n, n, i, j, binomial(i-1, j-1))); vector(n, k, (P*M)[n, k]);} \\ Michel Marcus, Feb 15 2022

Formula

P * M, as infinite lower triangular matrices. P = Pascal's triangle, M = mu(n) in the main diagonal and the rest zeros.

Extensions

More terms from Michel Marcus, Feb 15 2022
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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