0,8

Temba Shonhiwa, A Generalization of the Euler and Jordan Totient Functions, Fib. Quart., 37 (1999), 67-76.

Comments from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 12 2007: Triangle begins

1

1 1

0 1 1

0 2 3 1

0 2 5 4 1

0 4 10 10 5 1

0 2 11 19 15 6 1

0 6 21 35 35 21 7 1

0 4 22 52 69 56 28 8 1

0 6 33 83 126 126 84 36 9 1

0 4 34 110 205 251 210 120 45 10 1

The inverse of the triangle is

1

-1 1

1 -1 1

-1 1 -3 1

1 -1 7 -4 1

-1 1 -15 10 -5 1

1 -1 31 -19 15 -6 1

-1 1 -63 28 -35 21 -7 1

1 -1 127 -28 71 -56 28 -8 1

-1 1 -255 1 -135 126 -84 36 -9 1

1 -1 511 80 255 -251 210 -120 45 -10 1

with row sums 1,0,1,-2,4,-9,22,-55,135,-319,721,...(cf. A038200).

A020921 := proc(n, k) option remember ; local divs ; if n <= 0 then 1 ; elif k > n then 0 ; else divs := numtheory[divisors](n) ; add(numtheory[mobius](op(i, divs))*binomial(n/op(i, divs), k), i=1..nops(divs)) ; fi ; end: nmax := 10 ; for row from 0 to nmax do for col from 0 to row do printf("%d, ", A020921(row, col)) ; od ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 12 2007

(PARI) T(n, k)=if(n<=0, k==0&n==0, sumdiv(n, d, moebius(d)*binomial(n/d, k)))

(Left-hand) columns include A000010, A102309. Row sums are essentially A027375.

Sequence in context: A006703 A133623 A065862 this_sequence A154720 A071501 A004572

Adjacent sequences: A020918 A020919 A020920 this_sequence A020922 A020923 A020924

nonn,tabl,nice,easy

Michael Somos