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A321548 a(n) = Sum_{d|n} (-1)^(d-1)*d^9.

%I A321548 #20 Nov 04 2022 10:47:37
%S A321548 1,-511,19684,-262655,1953126,-10058524,40353608,-134480383,387440173,
%T A321548 -998047386,2357947692,-5170101020,10604499374,-20620693688,
%U A321548 38445332184,-68853957119,118587876498,-197981928403,322687697780,-512998309530,794320419872,-1204911270612,1801152661464,-2647111858972
%N A321548 a(n) = Sum_{d|n} (-1)^(d-1)*d^9.
%H A321548 Seiichi Manyama, <a href="/A321548/b321548.txt">Table of n, a(n) for n = 1..10000</a>
%H A321548 J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H A321548 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321548 G.f.: Sum_{k>=1} (-1)^(k-1)*k^9*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Dec 23 2018
%F A321548 Multiplicative with a(2^e) = 2 - (2^(9*e + 9) - 1)/511, and a(p^e) = (p^(9*e + 9) - 1)/(p^9 - 1) for p > 2. - _Amiram Eldar_, Nov 04 2022
%t A321548 Table[Total[(-1)^(#-1) #^9&/@Divisors[n]],{n,30}] (* _Harvey P. Dale_, Sep 07 2020 *)
%t A321548 f[p_, e_] := (p^(9*e + 9) - 1)/(p^9 - 1); f[2, e_] := 2 - (2^(9*e + 9) - 1)/511; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Nov 04 2022 *)
%o A321548 (PARI) apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^9), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321548 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%K A321548 sign,mult
%O A321548 1,2
%A A321548 _N. J. A. Sloane_, Nov 23 2018