A000743 - OEIS

%I M3852 N1577 #41 Jan 05 2025 19:51:31

%S 1,5,15,35,70,125,210,325,495,700,1000,1330,1820,2305,3060,3750,4830,

%T 5775,7315,8490,10625,12155,14880,16835,20475,22620,27405,30100,35750,

%U 39100,46360,49655,58905,62985,73320,78340,91390,95720,111930,117425

%N Number of compositions of n into 5 ordered relatively prime parts.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Marius A. Burtea, <a href="/A000743/b000743.txt">Table of n, a(n) for n = 5..10000</a>

%H H. W. Gould, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-4/gould.pdf">Binomial coefficients, the bracket function and compositions with relatively prime summands</a>, Fib. Quart. 2(4) (1964), 241-260.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Möbius transform of binomial(n-1, 4).

%F G.f.: Sum_{k>=1} mu(k) * x^(5*k) / (1 - x^k)^5. - _Ilya Gutkovskiy_, Feb 05 2020

%p with(numtheory):

%p a:= n-> add(mobius(n/d)*binomial(d-1, 4), d=divisors(n)):

%p seq(a(n), n=5..50);  # _Alois P. Heinz_, Feb 05 2020

%t a[n_] := Sum[Boole[Divisible[n, k]] MoebiusMu[n/k] Binomial[k - 1, 4], {k, 1, n}]; Table[a[n], {n, 5, 52}] (* _Jean-François Alcover_, Feb 11 2016 *)

%o (Magma) [&+[MoebiusMu(n div d)*Binomial(d-1, 4):d in Divisors(n)]:n in[5..44]]; // _Marius A. Burtea_, Feb 08 2020

%Y Cf. A000741, A000742, A023031, A023032, A023033, A023034, A023035.

%K nonn

%O 5,2

%A _N. J. A. Sloane_

%E Offset changed to 5 by _Ilya Gutkovskiy_, Feb 05 2020