A000742 Number of compositions of n into 4 ordered relatively prime parts.

1, 4, 10, 20, 34, 56, 80, 120, 154, 220, 266, 360, 420, 560, 614, 816, 884, 1120, 1210, 1540, 1572, 2020, 2080, 2544, 2638, 3276, 3200, 4060, 4040, 4840, 4896, 5960, 5710, 7140, 6954, 8216, 8136, 9880, 9244, 11480, 11010, 12824, 12650, 15180, 14024, 17276

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Marius A. Burtea, Table of n, a(n) for n = 4..10000
• H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000741, A000743, A023031, A023032, A023033, A023034, A023035.

Programs

Magma

[&+[MoebiusMu(n div d)*Binomial(d-1, 3):d in Divisors(n)]:n in[4..49]]; // Marius A. Burtea, Feb 08 2020

Maple

with(numtheory):
a:= n-> add(mobius(n/d)*binomial(d-1, 3), d=divisors(n)):
seq(a(n), n=4..50);  # Alois P. Heinz, Feb 05 2020

Mathematica

a[n_] := Sum[Boole[Divisible[n, k]] MoebiusMu[n/k] Binomial[k - 1, 3], {k, 1, n}]; Table[a[n], {n, 4, 51}] (* Jean-François Alcover, Feb 11 2016 *)

Formula

Möbius transform of C(n-1,3).

G.f.: Sum_{k>=1} mu(k) * x^(4*k) / (1 - x^k)^4. - Ilya Gutkovskiy, Feb 05 2020

Extensions

Offset changed to 4 by Ilya Gutkovskiy, Feb 05 2020