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.

A127513 Partial sums of A127511.

Original entry on oeis.org

1, -1, -5, -5, -21, 11, -53, -53, -53, 459, -565, -565, -4661, 3531, 19915, 19915, -45621, -45621, -307765, -307765, 740811, 2837963, -1356341, -1356341, -1356341, 32198091, 32198091, 32198091, -236237365, -773108277
Offset: 1

Views

Author

Gary W. Adamson, Jan 17 2007

Keywords

Examples

			a(3) = -5 = (1 - 2 - 4).

Links

Crossrefs

Cf. A127511, A127512.

Programs

  • Maple
    a:= proc(n) option remember;
      numtheory[mobius](n)*2^(n-1) +`if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Apr 04 2012
  • Mathematica
    Table[2^(n-1) MoebiusMu[n], {n, 1, 30}] // Accumulate (* Jean-François Alcover, May 21 2020 *)

Extensions

More terms from R. J. Mathar, Apr 04 2012

A127512 Triangle read by rows: T(n,k)= mobius(n)*binomial(n-1,k-1).

Original entry on oeis.org

1, -1, -1, -1, -2, -1, 0, 0, 0, 0, -1, -4, -6, -4, -1, 1, 5, 10, 10, 5, 1, -1, -6, -15, -20, -15, -6, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, -1, -10, -45, -120, -210, -252, -210, -120, -45, -10, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gary W. Adamson, Jan 17 2007

Keywords

Comments

Could also be defined as the matrix product of A128407 and A007318.
A013929 gives the indices of rows that are all zeros. - Michel Marcus, Feb 15 2022

Examples

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

Links

Crossrefs

Cf. A007318, A008683, A013929, A127511 (row sums).
Cf. A127514 (P*M).

Programs

  • Maple
    A127512 := proc(n,k)
    numtheory[mobius](n)*binomial(n-1,k-1) ;
end proc:
seq(seq( A127512(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Aug 15 2022
  • Mathematica
    T[n_,k_]:= MoebiusMu[n]*Binomial[n-1,k-1];Table[T[n,k],{n,12},{k,n}]//Flatten (* James C. McMahon, Jan 02 2025 *)
  • 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, (M*P)[n, k]); \\ Michel Marcus, Feb 15 2022

Formula

T(n,k) = mu(n)*binomial(n-1,k-1) = A008683(n)*A007318(n-1,k-1). - R. J. Mathar, Aug 15 2022

Extensions

Edited by N. J. A. Sloane, Sep 25 2008
NAME simplified by R. J. Mathar, Aug 15 2022

A156734 Square array read by antidiagonals up. T(n,k) = if k divides n then +1 else -1.

Original entry on oeis.org

1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, 1, -1, -1, 1, -1, -1, -1, -1, -1, 1, 1, -1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1
Offset: 1

Views

Author

Mats Granvik, Feb 14 2009

Keywords

Comments

Swap the element in the lower right corner with the element in the upper right corner and calculate the determinant. The result is sequence A127511.

Examples

			Table begins:
1.-1.-1.-1.-1.-1.-1.-1.-1.-1.-1.-1.-1
1..1.-1.-1.-1.-1.-1.-1.-1.-1.-1.-1.-1
1.-1..1.-1.-1.-1.-1.-1.-1.-1.-1.-1.-1
1..1.-1..1.-1.-1.-1.-1.-1.-1.-1.-1.-1
1.-1.-1.-1..1.-1.-1.-1.-1.-1.-1.-1.-1
1..1..1.-1.-1..1.-1.-1.-1.-1.-1.-1.-1
1.-1.-1.-1.-1.-1..1.-1.-1.-1.-1.-1.-1
1..1.-1..1.-1.-1.-1..1.-1.-1.-1.-1.-1
1.-1..1.-1.-1.-1.-1.-1..1.-1.-1.-1.-1
1..1.-1.-1..1.-1.-1.-1.-1..1.-1.-1.-1
1.-1.-1.-1.-1.-1.-1.-1.-1.-1..1.-1.-1
1..1..1..1.-1..1.-1.-1.-1.-1.-1..1.-1
1.-1.-1.-1.-1.-1.-1.-1.-1.-1.-1.-1..1

Crossrefs

Cf. A127511.

Programs

  • Excel
    =if(or(mod(row();column())=0);1;-1)

Formula

T(n,k) = if k divides n then +1 else -1.
Showing 1-3 of 3 results.

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

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