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A127512
Triangle read by rows: T(n,k)= mobius(n)*binomial(n-1,k-1).
4
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
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
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
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