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

Author

Gary W. Adamson, Jan 17 2007

Keywords

• tabl
• sign
• easy

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

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

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