A128407 -id:A128407 - cp's OEIS frontend

A143469 Triangle read by rows, A000012 * A143315 * A128407, 1<=k<=1.

1, 4, -1, 9, -1, -1, 16, -4, -1, 0, 25, -4, -1, 0, -1, 36, -9, -4, 0, -1, 1, 49, -9, -4, 0, -1, 1, -1, 64, -16, -4, 0, -1, 1, -1, 0, 81, -16, -9, 0, -1, 1, -1, 0, 0, 100, -25, -9, 0, -4, 1, -1, 0, 0, 1, 121, -25, -9, 0, -4, 1, -1, 0, 0, 1, -1, 144, -36, -16, 0, -4, 4, -1, 0, 0, 1, -1, 0

Offset: 1

Gary W. Adamson, Aug 17 2008

Row sums = A018805: (1, 3, 7, 11, 19, 23, 35,...).

Right border = mu(n), A008683.

			First few rows of the triangle =
1;
4, -1;
9, -1, -1;
16, -4, -1, 0;
25, -4, -1, 0, -1;
36, -9, -4, 0, -1, 1;
49, -9, -4, 0, -1, 1, -1;
...

Cf. A008683, A018805, A128407, A143315.

a(9) corrected and more terms from Georg Fischer, Aug 14 2023

A128431 Triangle read by rows: A054521 * A128407.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 03 2007

Row sums = A112399: (1, 1, 0, 0, -1, 0, -1, -2, -2, -1, ...).

			First few rows of the triangle:
  1;
  1,  0;
  1, -1,  0;
  1,  0, -1,  0;
  1, -1, -1,  0,  0;
  1,  0,  0,  0, -1,  0;
  1, -1, -1,  0, -1,  1,  0;

Cf. A054521, A128407, A112399, A008683.

A054521 * A128407 as infinite lower triangular matrices.

A128432 Triangle read by rows: A128407 * A054521.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 03 2007

Row sums = A023900: (1, -1, -2, 0, -4, 2, -6, ...) Left border = mu(n), A008683.

			First few rows of the triangle:
   1;
  -1,  0;
  -1, -1,  0;
   0,  0,  0,  0;
  -1, -1, -1, -1,  0;
   1,  0,  0,  0,  1,  0;
  -1, -1, -1, -1, -1, -1,  0;
   0,  0,  0,  0,  0,  0,  0,  0;
  ...

Cf. A128407, A054521, A008683, A023900.

A128407 * A054521 as infinite lower triangular matrices.

A143352 Triangle read by rows, A051731 * A054524 = (A051731)^2 * A128407; 1<=k<=n.

Original entry on oeis.org

1, 2, -1, 2, 0, -1, 3, -2, 0, 0, 2, 0, 0, 0, -1, 4, -2, -2, 0, 0, 1, 2, 0, 0, 0, 0, 0, -1, 4, -3, 0, 0, 0, 0, 0, 0, 3, 0, -2, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, -2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 6, -4, -3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 4, -2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Aug 10 2008

Left border = d(n), A000010.
Right border = mu(n), A008683.
Row sums = 1.

			First few rows of the triangle =
1;
2, -1;
2, 0, -1;
3, -2, 0, 0;
2, 0, 0, 0, -1
4, -2, -2, 0, 0, 1;
2, 0, 0, 0, 0, 0, -1;
4, -3, 0, 0, 0, 0, 0, 0;
3, 0, -2, 0, 0, 0, 0, 0, 0;
4, -2, 0, 0, -2, 0, 0, 0, 0, 1;
...

Cf. A054524, A051731, A128407, A000010, A008683.

Triangle read by rows, A051731 * A054524 = (A051731)^2 * A128407; 1<=k<=n

A054524 Triangle T(n,k): T(n,k) = mu(k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1<=k<=n).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 09 2000

Cf. A054521.

Cf. A051731, A128407.

Clear[t]; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, 1, If[n == k, -Sum[t[n, k - i], {i, 1, k - 1}], If[n > k, t[n - k, k], 0]]]; Flatten[Table[t[n, k], {n, 13}, {k, n}]] (* Mats Granvik, Feb 12 2012 *)
Table[If[Mod[n,k]==0,MoebiusMu[k],0],{n,20},{k,n}]//Flatten (* Harvey P. Dale, Mar 29 2023 *)

A054524 = A051731 * A128407 - Gary W. Adamson, Mar 02 2007

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

Gary W. Adamson, Jan 17 2007

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

			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;
  ...

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A007318, A008683, A013929, A127511 (row sums).

Cf. A127514 (P*M).

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

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 *)

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

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

Edited by N. J. A. Sloane, Sep 25 2008

NAME simplified by R. J. Mathar, Aug 15 2022

A128409 Triangle read by rows: A000012 * A128408 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, -1, -1, -1, -1, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, 0, 0, 0, -1, 1, -2, 0, 0, 0, -1, 1, -1, -2, 0, 0, 0, -1, 1, -1, 0, -2, 0, 0, 0, -1, 1, -1, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 1, -2, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, -2, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, -3, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, -1

Offset: 1

Gary W. Adamson, Mar 01 2007

Left border = the Mertens sequence: A002321: (1, 0, -1, -1, -2, ...).
Right border = mu(n), A008683: (1, -1, -1, 0, -1, 1, -1, ...).
Row sums = A062563: (1, -1, -3, -3, -5, -1, -3, ...).

			First few rows of the triangle:
   1;
   0, -1;
  -1, -1, -1;
  -1, -1, -1,  0;
  -2, -1, -1,  0, -1;
  -1,  0,  0,  0, -1,  1;
  -2,  0,  0,  0, -1,  1, -1;
  ...

Cf. A002321, A008683, A062563, A128407, A128408.

Previous a(51) = 0 removed and more terms from Georg Fischer, Jun 08 2023

A143728 Triangle read by rows: termwise product of mu(n) and n-th row of A127368.

Original entry on oeis.org

1, 1, 0, 1, -2, 0, 1, 0, -3, 0, 1, -2, -3, 0, 0, 1, 0, 0, 0, -5, 0, 1, -2, -3, 0, -5, 6, 0, 1, 0, -3, 0, -5, 0, -7, 0, 1, -2, 0, 0, -5, 0, -7, 0, 0, 1, 0, -3, 0, 0, 0, -7, 0, 0, 0, 1, -2, -3, 0, -5, 6, -7, 0, 0, 10, 0, 1, 0, 0, 0, -5, 0, -7, 0, 0, 0, -11, 0, 1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -11, 0, 0

Offset: 1

Gary W. Adamson, Aug 30 2008

The operation A127368 * A128407 forms the termwise product of mu(n) and the n-th row of A127368: deleting all squares and changing the sign of primes to (-1).

Row sums = A143729: (1, 1, -1, -2, -4, -4, -3, -14, ...)

			First few terms of the triangle:
  1;
  1,  0;
  1, -2,  0;
  1,  0, -3,  0;
  1, -2, -3,  0,  0;
  1,  0,  0,  0, -5,  0;
  1, -2, -3,  0, -5,  6,  0;
  1,  0, -3,  0, -5,  0, -7,  0;
  ...
Example: row 7 = (1, -2, -3, 0, -5, 6, 0). We take row 7 of triangle A127368 which records the relative primes of 7 as (1, 2, 3, 4, 5, 6, 0). Applying the termwise product of the first 7 terms of mu(k): (1, -1, -1, 0, -1, 1, -1), we get (1, -2, -3, 0, -5, 6, 0), noting that the "4" has been deleted.

Cf. A008683, A127368, A128407.

Triangle read by rows, A127368 * A128407, 1 <= k <= n; T(n,k) = {1<=k<=n, gcd(k,n)=1} * mu(k).

Partially edited by N. J. A. Sloane, Jan 05 2009

a(66) = 0 inserted by Georg Fischer, Jun 05 2023

A128410 A128408 * A000012.

Original entry on oeis.org

1, -2, -1, -2, -1, -1, 0, 0, 0, 0, -2, -1, -1, -1, -1, 4, 3, 2, 1, 1, 1, -2, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Mar 01 2007

Left border = A008966: (1, -2, -2, 0, -2, 4, -2, 0, 0, 4, ...). Row sums = A063441: (1, -3, -4, 0, -6, 12, ...) A128409 = A000012 * A128408.

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

Cf. A128407, A128408, A128409, A008966, A051731, A063441.

A128408 * A000012 as infinite lower triangular matrices. Partial row sums of A128408 starting from the right.

A128430 Triangle read by rows: A054524 * A000012.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 02 2007

Right border = mu(n), A008683. Row sums = A023900: (1, -1, -2, -1, -4, 2, -6, -1, ...). A054524 = A051731 * A128407.

			First few rows of the triangle:
  1;
  0, -1;
  0, -1, -1;
  0, -1,  0,  0;
  0, -1, -1, -1, -1;
  0, -1,  0,  1,  1,  1;
  0, -1, -1, -1, -1, -1, -1;
  0, -1,  0,  0,  0,  0,  0,  0;
  ...

Cf. A054524, A000012, A051731, A128407, A008683, A023900.

A054524 * A000012 as infinite lower triangular matrices.

cp's OEIS Frontend

A143469 Triangle read by rows, A000012 * A143315 * A128407, 1<=k<=1.

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A128431 Triangle read by rows: A054521 * A128407.

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A128432 Triangle read by rows: A128407 * A054521.

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A143352 Triangle read by rows, A051731 * A054524 = (A051731)^2 * A128407; 1<=k<=n.

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A054524 Triangle T(n,k): T(n,k) = mu(k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1<=k<=n).

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

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A128409 Triangle read by rows: A000012 * A128408 as infinite lower triangular matrices.

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A143728 Triangle read by rows: termwise product of mu(n) and n-th row of A127368.

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A128410 A128408 * A000012.

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