A320043 - OEIS

%I #43 Sep 08 2022 08:46:23

%S 1,6,13,50,37,196,189,384,351,1210,601,2366,1471,2156,2941,6936,3277,

%T 10830,5563,9022,9681,23276,9897,26300,19267,30030,23043,58870,21087,

%U 76880,46717,59296,57801,83546,50281,156066,90973,117968,90539,235340,86179,284746

%N Row sums of the triangle A322550.

%C Conjecture: a(n) is not a perfect square except for n = 1, 6 and 96.

%H Stefano Spezia, <a href="/A320043/b320043.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} (n + 1 - k)^2*k/gcd(n + 1 - k, k)^3.

%F a(n) = Sum_{k=1..n} A000290(n + 1 - k)*A000027(k)/A000578(A050873(n + 1 - k, k)).

%p a := n -> sum((n+1-k)^2*k/gcd(n+1-k, k)^3, k = 1 .. n): seq(a(n), n = 1 .. 50);

%t a[n_]:=Sum[(n+1-k)^2*k/GCD[n+1-k,k]^3, {k, 1, n}]; Array[a, 50]

%o (GAP) List([1..50], n->Sum([1..n], k->(n+1-k)^2*k/GcdInt(n+1-k,k)^3));

%o (Magma) [(&+[(n+1-k)^2*k/Gcd(n+1-k,k)^3: k in [1..n]]): n in [1..50]];

%o (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 (PARI)

%o a(n) = sum(k=1, n, (n+1-k)^2*k/gcd(n+1-k,k)^3);

%o vector(50, n, a(n))

%Y Cf. A000027, A000290, A000578, A050873, A322550.

%K nonn

%O 1,2

%A _Stefano Spezia_, Dec 16 2018