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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.

%I A020921 #36 Jan 05 2025 19:51:34
%S A020921 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 A020921 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 A020921 34,110,205,251,210,120,45,10,1,0,10,55,165,330,462,462,330,165,55,11
%N 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.
%H A020921 Temba Shonhiwa, <a href="https://web.archive.org/web/2024*/https://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 A020921 From _R. J. Mathar_, Feb 12 2007: (Start)
%e A020921 Triangle begins
%e A020921   1
%e A020921   1 1
%e A020921   0 1  1
%e A020921   0 2  3   1
%e A020921   0 2  5   4   1
%e A020921   0 4 10  10   5   1
%e A020921   0 2 11  19  15   6   1
%e A020921   0 6 21  35  35  21   7   1
%e A020921   0 4 22  52  69  56  28   8  1
%e A020921   0 6 33  83 126 126  84  36  9  1
%e A020921   0 4 34 110 205 251 210 120 45 10 1
%e A020921 The inverse of the triangle is
%e A020921    1
%e A020921   -1    1
%e A020921    1   -1    1
%e A020921   -1    1   -3    1
%e A020921    1   -1    7   -4    1
%e A020921   -1    1  -15   10   -5    1
%e A020921    1   -1   31  -19   15   -6    1
%e A020921   -1    1  -63   28  -35   21   -7    1
%e A020921    1   -1  127  -28   71  -56   28   -8    1
%e A020921   -1    1 -255    1 -135  126  -84   36   -9    1
%e A020921    1   -1  511   80  255 -251  210 -120   45  -10    1
%e A020921 with row sums 1,0,1,-2,4,-9,22,-55,135,-319,721,...(cf. A038200).
%e A020921 (End)
%p A020921 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 A020921 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 A020921 (PARI) {T(n, k) = if( n<=0, k==0 && n==0, sumdiv(n, d, moebius(d) * binomial(n/d, k)))}
%o A020921 (Sage) # uses[DivisorTriangle from A327029]
%o A020921 DivisorTriangle(moebius, binomial, 13) # _Peter Luschny_, Aug 24 2019
%Y A020921 (Left-hand) columns include A000010, A102309. Row sums are essentially A027375.
%Y A020921 Cf. A327029.
%K A020921 nonn,tabl,nice,easy
%O A020921 0,8
%A A020921 _Michael Somos_, Nov 17 2002