A127512 - OEIS

%I #37 Jan 02 2025 10:50:11

%S 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,

%T -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,

%U 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

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

%C Could also be defined as the matrix product of A128407 and A007318.

%C A013929 gives the indices of rows that are all zeros. - _Michel Marcus_, Feb 15 2022

%H James C. McMahon, <a href="/A127512/b127512.txt">Table of n, a(n) for n = 1..10000</a>

%F T(n,k) = mu(n)*binomial(n-1,k-1) = A008683(n)*A007318(n-1,k-1). - _R. J. Mathar_, Aug 15 2022

%e First few rows of the triangle:

%e    1;

%e   -1, -1;

%e   -1, -2, -1;

%e    0,  0,  0,  0;

%e   -1, -4, -6, -4, -1;

%e    1,  5, 10, 10,  5, 1;

%e   ...

%p A127512 := proc(n,k)

%p     numtheory[mobius](n)*binomial(n-1,k-1) ;

%p end proc:

%p seq(seq( A127512(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Aug 15 2022

%t 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 *)

%o (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

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

%Y Cf. A127514 (P*M).

%K tabl,sign,easy

%O 1,5

%A _Gary W. Adamson_, Jan 17 2007

%E Edited by _N. J. A. Sloane_, Sep 25 2008

%E NAME simplified by _R. J. Mathar_, Aug 15 2022