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

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

1,5

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

Links

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

Formula

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

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

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

Cf. A127514 (P*M).

Sequence in context: A308118 A017857 A127842 * A263787 A307710 A112207

Adjacent sequences: A127509 A127510 A127511 * A127513 A127514 A127515

Keyword

tabl,sign,easy

Author

Gary W. Adamson, Jan 17 2007

Extensions

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

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

Status

approved