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%I #34 Mar 24 2020 12:37:41
%S 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,
%T 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,
%U 34,110,205,251,210,120,45,10,1,0,10,55,165,330,462,462,330,165,55,11
%N 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.
%H Temba Shonhiwa, <a href="http://www.fq.math.ca/Scanned/37-1/shonhiwa.pdf">A Generalization of the Euler and Jordan Totient Functions</a>, Fib. Quart., 37 (1999), 67-76.
%e From _R. J. Mathar_, Feb 12 2007: (Start)
%e Triangle begins
%e 1
%e 1 1
%e 0 1 1
%e 0 2 3 1
%e 0 2 5 4 1
%e 0 4 10 10 5 1
%e 0 2 11 19 15 6 1
%e 0 6 21 35 35 21 7 1
%e 0 4 22 52 69 56 28 8 1
%e 0 6 33 83 126 126 84 36 9 1
%e 0 4 34 110 205 251 210 120 45 10 1
%e The inverse of the triangle is
%e 1
%e -1 1
%e 1 -1 1
%e -1 1 -3 1
%e 1 -1 7 -4 1
%e -1 1 -15 10 -5 1
%e 1 -1 31 -19 15 -6 1
%e -1 1 -63 28 -35 21 -7 1
%e 1 -1 127 -28 71 -56 28 -8 1
%e -1 1 -255 1 -135 126 -84 36 -9 1
%e 1 -1 511 80 255 -251 210 -120 45 -10 1
%e with row sums 1,0,1,-2,4,-9,22,-55,135,-319,721,...(cf. A038200).
%e (End)
%p 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
%t 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 *)
%o (PARI) {T(n, k) = if( n<=0, k==0 && n==0, sumdiv(n, d, moebius(d) * binomial(n/d, k)))}
%o (Sage) # uses[DivisorTriangle from A327029]
%o DivisorTriangle(moebius, binomial, 13) # _Peter Luschny_, Aug 24 2019
%Y (Left-hand) columns include A000010, A102309. Row sums are essentially A027375.
%Y Cf. A327029.
%K nonn,tabl,nice,easy
%O 0,8
%A _Michael Somos_, Nov 17 2002
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