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