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A320043 Row sums of the triangle A322550.

%I A320043 #43 Sep 08 2022 08:46:23
%S A320043 1,6,13,50,37,196,189,384,351,1210,601,2366,1471,2156,2941,6936,3277,
%T A320043 10830,5563,9022,9681,23276,9897,26300,19267,30030,23043,58870,21087,
%U A320043 76880,46717,59296,57801,83546,50281,156066,90973,117968,90539,235340,86179,284746
%N A320043 Row sums of the triangle A322550.
%C A320043 Conjecture: a(n) is not a perfect square except for n = 1, 6 and 96.
%H A320043 Stefano Spezia, <a href="/A320043/b320043.txt">Table of n, a(n) for n = 1..10000</a>
%F A320043 a(n) = Sum_{k=1..n} (n + 1 - k)^2*k/gcd(n + 1 - k, k)^3.
%F A320043 a(n) = Sum_{k=1..n} A000290(n + 1 - k)*A000027(k)/A000578(A050873(n + 1 - k, k)).
%p A320043 a := n -> sum((n+1-k)^2*k/gcd(n+1-k, k)^3, k = 1 .. n): seq(a(n), n = 1 .. 50);
%t A320043 a[n_]:=Sum[(n+1-k)^2*k/GCD[n+1-k,k]^3, {k, 1, n}]; Array[a, 50]
%o A320043 (GAP) List([1..50], n->Sum([1..n], k->(n+1-k)^2*k/GcdInt(n+1-k,k)^3));
%o A320043 (Magma) [(&+[(n+1-k)^2*k/Gcd(n+1-k,k)^3: k in [1..n]]): n in [1..50]];
%o A320043 (Maxima) a(n):=sum((n+1-k)^2*k/gcd(n+1-k,k)^3, k, 1, n)$ makelist(a(n), n, 1, 50);
%o A320043 (PARI)
%o A320043 a(n) = sum(k=1, n, (n+1-k)^2*k/gcd(n+1-k,k)^3);
%o A320043 vector(50, n, a(n))
%Y A320043 Cf. A000027, A000290, A000578, A050873, A322550.
%K A320043 nonn
%O A320043 1,2
%A A320043 _Stefano Spezia_, Dec 16 2018