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A343544 a(n) = n * Sum_{d|n} binomial(d+2,3)/d.

%I A343544 #20 Apr 26 2021 13:02:30
%S A343544 1,6,13,32,40,94,91,184,204,320,297,612,468,770,850,1184,986,1752,
%T A343544 1349,2280,2114,2662,2323,4184,3125,4264,4266,5740,4524,7660,5487,
%U A343544 8352,7546,9180,8470,13212,9176,12654,12194,16640,12382,19628,14233,20724,19590,22034,18471,30416,21462
%N A343544 a(n) = n * Sum_{d|n} binomial(d+2,3)/d.
%H A343544 Robert Israel, <a href="/A343544/b343544.txt">Table of n, a(n) for n = 1..10000</a>
%F A343544 G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^4 = Sum_{k>=1} binomial(k+2,3) * x^k/(1 - x^k)^2.
%p A343544 f:= n -> n/6*add((d+1)*(d+2),d=numtheory:-divisors(n)):
%p A343544 map(f, [$1..100]); # _Robert Israel_, Apr 26 2021
%t A343544 a[n_] := n * DivisorSum[n, Binomial[# + 2, 3]/# &]; Array[a, 50] (* _Amiram Eldar_, Apr 25 2021 *)
%o A343544 (PARI) a(n) = n*sumdiv(n, d, binomial(d+2, 3)/d);
%o A343544 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+2, 3)*x^k/(1-x^k)^2))
%Y A343544 Cf. A000203, A038040, A059358, A309731, A343545, A343546, A343547.
%K A343544 nonn
%O A343544 1,2
%A A343544 _Seiichi Manyama_, Apr 19 2021