A077050 -id:A077050 - cp's OEIS frontend

A054525 Triangle T(n,k): T(n,k) = mu(n/k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

1, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 1, -1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

N. J. A. Sloane, Apr 09 2000

A051731 = the inverse of this triangle = A129372 * A115361. - Gary W. Adamson, Apr 15 2007

If a column T(n,0)=0 is added, these are the coefficients of the necklace polynomials multiplied by n [Moree, Metropolis]. - R. J. Mathar, Nov 11 2008

			Triangle (with rows n >= 1 and columns k >= 1) begins as follows:
   1;
  -1,  1;
  -1,  0,  1;
   0, -1,  0,  1;
  -1,  0,  0,  0,  1;
   1, -1, -1,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   0,  0,  0, -1,  0,  0,  0,  1; ...
Matrix inverse is triangle A051731:
  1;
  1, 1;
  1, 0, 1;
  1, 1, 0, 1;
  1, 0, 0, 0, 1;
  1, 1, 1, 0, 0, 1;
  1, 0, 0, 0, 0, 0, 1;
  1, 1, 0, 1, 0, 0, 0, 1; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows
Trevor Hyde, Cyclotomic factors of necklace polynomials, arXiv:1811.08601 [math.CO], 2018.
N. Metropolis and G.-C. Rota, Witt vectors and the algebra of necklaces, Adv. Math. 50 (1983), 95-125.
Pieter Moree, The formal series Witt transform, Discr. Math. 295 (2005), 143-160.

Cf. A054521.
Cf. A051731, A115361, A129372.
Cf. A077050, A115359, A129360.

A054525 := proc(n,k)
    if n mod k = 0 then
        numtheory[mobius](n/k) ;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Oct 21 2012

t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k ], 0]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

tabl(nn) = {T = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); for (n=1, nn, for (k=1, n, print1(T[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

row(n) = Vecrev(sumdiv(n, d, moebius(d)*x^(n/d))/x); \\ Michel Marcus, Aug 24 2021

from math import isqrt, comb
from sympy import mobius
def A054525(n): return 0 if (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))%(b:=n-comb(a,2)) else mobius(a//b) # Chai Wah Wu, Nov 13 2024

Matrix inverse of triangle A051731, where A051731(n, k) = 1 if k|n, 0 otherwise. - Paul D. Hanna, Jan 09 2006
Equals = A129360 * A115359 as infinite lower triangular matrices. - Gary W. Adamson, Apr 15 2007
Bivariate g.f.: Sum_{n, k >= 1} T(n, k)*x^n*y^k = Sum_{m >= 1} mu(m)*x^m*y/(1 - x^m*y). - Petros Hadjicostas, Jun 25 2019

A077049 Left summatory matrix, T, by antidiagonals upwards.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Oct 22 2002

If S = (s(1), s(2), ...) is a sequence written as a column vector, then T*S is the summatory sequence of S; i.e., its n-th term is Sum_{k|n} s(k). T is the inverse of the left Moebius transformation matrix, A077050. Except for the first term in some cases, column 1 of T^(-2) is A007427, column 1 of T^(-1) is A008683, Column c of T^2 is A000005, column 1 of T^3 is A007425.
This is essentially the same as A051731, which includes only the triangle. Note that the standard in the OEIS is left to right antidiagonals, which would make this the right summatory matrix, and A077051 the left one. - Franklin T. Adams-Watters, Apr 08 2009
From Gary W. Adamson, Apr 28 2010: (Start)
As defined with antidiagonals of the array = the triangle shown in the example section. Row sums of this triangle = A032741 (with a different offset): 1, 1, 2, 1, 3, 1, 3, ...
Let the triangle = M. Then lim_{n->inf} M^n = A002033, the left-shifted vector considered as a sequence: (1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, ...). (End)

			T(4,2) = 1 since 2 divides 4. Northwest corner:
  1 0 0 0 0 0
  1 1 0 0 0 0
  1 0 1 0 0 0
  1 1 0 1 0 0
  1 0 0 0 1 0
  1 1 1 0 0 1
From _Gary W. Adamson_, Apr 28 2010: (Start)
First few rows of the triangle (when T is read by antidiagonals upwards):
  1;
  1, 0;
  1, 1, 0;
  1, 0, 0, 0;
  1, 1, 1, 0, 0;
  1, 0, 0, 0, 0, 0;
  1, 1, 0, 1, 0, 0, 0;
  1, 0, 1, 0, 0, 0, 0, 0;
  1, 1, 0, 0, 1, 0, 0, 0, 0;
  ... (End)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (Rows 1 <= n <= 150).
Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

Cf. A051731, A077050, A077051, A077052, A000005 (row sums).

Cf. A032741, A002033. - Gary W. Adamson, Apr 28 2010

A077049 := proc(n,k)
    if modp(n,k) = 0 then
        1;
    else
        0 ;
    end if;
end proc:
for d from 2 to 10 do
    for k from 1 to d-1 do
        n := d-k ;
        printf("%d,",A077049(n,k)) ;
    end do:
end do: # R. J. Mathar, Jul 22 2017

With[{nn = 14}, DeleteCases[#, -1] & /@ Transpose@ Table[Take[#, nn] &@ Flatten@ Join[ConstantArray[-1, k - 1], ConstantArray[Reverse@ IntegerDigits[2^(k - 1), 2], Ceiling[(nn - k + 1)/k]]], {k, nn}]] // Flatten (* Michael De Vlieger, Jul 22 2017 *)

nn=10; matrix(nn, nn, n, k, if (n % k, 0, 1)) \\ Michel Marcus, May 21 2015

def T(n, k):
    return 1 if n%k==0 else 0
for n in range(1, 11): print([T(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jul 22 2017

T(n,k)=1 if k|n, otherwise T(n,k)=0, k >= 1, n >= 1.
From Boris Putievskiy, May 08 2013: (Start)
As table T(n,k) = floor(k/n) - floor((k-1)/n).
As linear sequence a(n) = floor(A004736(n)/A002260(n)) - floor((A004736(n)-1)/A002260(n)); a(n) = floor(j/i)-floor((j-1)/i), where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

Name edited by Petros Hadjicostas, Jul 27 2019

A077051 Right summatory matrix, T, by antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1

Offset: 1

Clark Kimberling, Oct 22 2002

If S=(s(1),s(2),...) is a sequence written as a row vector, then S*T is the summatory sequence of S; i.e. its n-th term is Sum{s(k): k|n}. T is the transpose of the left summatory matrix, A077049; T is the inverse of the right Moebius transformation matrix. See A077049 for further properties.
This is essentially the same as A051731, which includes only the triangle. Note that the standard in the OEIS is left to right antidiagonals, which would make this the left summatory matrix, and A077049 the right one. - Franklin T. Adams-Watters, Apr 08 2009
Sum of column k is A000005. - Geoffrey Critzer, Mar 29 2015

			Northwest corner:
1 1 1 1 1 1
0 1 0 1 0 1
0 0 1 0 0 1
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1

Cf. A077049, A077050, A077052.

(* returns the northwest corner *) nn = 20; Table[PadRight[Drop[CoefficientList[Series[x^n/(1 - x^n), {x, 0, nn}], x],1], nn], {n, 1, nn}] // Grid (* Geoffrey Critzer, Mar 29 2015 *)

T(n, k)=1 if n|k else T(n, k)=0.
From Boris Putievskiy, May 08 2013: (Start)
As table T(n,k)= floor(n/k)-floor((n-1)/k).
As linear sequence a(n) = floor(A002260(n)/A004736(n)) - floor((A002260(n)-1)/A004736(n));
a(n) = floor(i/j) - floor((i-1)/j), where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)
G.f. for row n: x^n/(1-x^n). - Geoffrey Critzer, Mar 29 2015

A077052 Right Moebius transformation matrix, M, by antidiagonals.

Original entry on oeis.org

1, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Oct 22 2002

If S=(s(1),s(2),...) is a sequence written as a row vector, then S*M is the Moebius transform of S; i.e. its n-th term is Sum{mu(k)*s(k): k|n}. M is the transpose of the left Moebius transformation matrix, A077050.

			Northwest corner:
1 -1 -1 0 -1 1
0 1 0 -1 0 -1
0 0 1 0 0 -1
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1

Cf. A077049, A077050, A077051.

M=T^(-1), where T is the right summatory matrix, A077051.

cp's OEIS Frontend

A054525 Triangle T(n,k): T(n,k) = mu(n/k) if k divides n, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

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A077049 Left summatory matrix, T, by antidiagonals upwards.

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A077051 Right summatory matrix, T, by antidiagonals.

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A077052 Right Moebius transformation matrix, M, by antidiagonals.

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