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A117109 Moebius transform of binomial(n+3, 4).

%I A117109 #21 Feb 01 2023 12:28:41
%S A117109 1,4,14,30,69,107,209,295,480,641,1000,1209,1819,2166,2976,3546,4844,
%T A117109 5379,7314,8110,10402,11645,14949,15890,20405,21927,26910,29055,35959,
%U A117109 37108,46375,48484,57890,61196,73536,75027,91389,93951,110096,114260
%N A117109 Moebius transform of binomial(n+3, 4).
%C A117109 Partial sums of a(n) give A015650(n).
%H A117109 Robert Israel, <a href="/A117109/b117109.txt">Table of n, a(n) for n = 1..10000</a>
%F A117109 a(n) = |{(x,y,z,w) : 1 <= x <= y <= z <= w <= n, gcd(x,y,z,w,n) = 1}|.
%F A117109 G.f.: Sum_{k>=1} mu(k) * x^k / (1 - x^k)^5. - _Ilya Gutkovskiy_, Feb 13 2020
%e A117109 a(2)=4 because of the quadruples (1,1,1,1), (1,1,1,2), (1,1,2,2), (1,2,2,2).
%p A117109 b34:= unapply(expand(binomial(n+3,4)),n):
%p A117109 f:= proc(n) local k; uses numtheory;
%p A117109 add(b34(k)*mobius(n/k),k=divisors(n))
%p A117109 end proc:
%p A117109 map(f, [$1..100]); # _Robert Israel_, May 24 2019
%t A117109 a[n_] := Sum[Binomial[k+3, 4] MoebiusMu[n/k], {k, Divisors[n]}];
%t A117109 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 01 2023 *)
%o A117109 (PARI) a(n) = sumdiv(n, k, binomial(k+3, 4)*moebius(n/k)); \\ _Michel Marcus_, Nov 04 2018
%Y A117109 Cf. A007438, A015650, A117108.
%K A117109 nonn
%O A117109 1,2
%A A117109 _Steve Butler_, Apr 18 2006