A020921 Triangle read by rows: T(m,n) = number of solutions to 1 <= a(1) < a(2) < ... < a(m) <= n, where gcd( a(1), a(2), ..., a(m), n) = 1.

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, 0, 10, 55, 165, 330, 462, 462, 330, 165, 55, 11

Offset

0,8

Links

Table of n, a(n) for n=0..76.

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

Example

From R. J. Mathar, Feb 12 2007: (Start)
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).
(End)

Maple

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, Feb 12 2007

Mathematica

nmax = 11; t[n_, k_] := Total[ MoebiusMu[#]*Binomial[n/#, k] & /@ Divisors[n]]; t[0, 0] = 1; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Oct 20 2011, after PARI *)

Prog

(PARI) {T(n, k) = if( n<=0, k==0 && n==0, sumdiv(n, d, moebius(d) * binomial(n/d, k)))}
(Sage) # uses[DivisorTriangle from A327029]
DivisorTriangle(moebius, binomial, 13) # Peter Luschny, Aug 24 2019

Crossrefs

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

Cf. A327029.

Sequence in context: A365196 A372257 A253580 * A293113 A366528 A154720

Adjacent sequences: A020918 A020919 A020920 * A020922 A020923 A020924

Keyword

nonn,tabl,nice,easy

Author

Michael Somos, Nov 17 2002

Status

approved