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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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)))}

CROSSREFS

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

Sequence in context: A065862 A189117 A253580 * A293113 A154720 A071501

Adjacent sequences:  A020918 A020919 A020920 * A020922 A020923 A020924

KEYWORD

nonn,tabl,nice,easy

AUTHOR

Michael Somos, Nov 17 2002

STATUS

approved

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Last modified May 21 14:57 EDT 2019. Contains 323443 sequences. (Running on oeis4.)