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A321809 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^12.

%I A321809 #18 Nov 22 2022 22:01:13
%S A321809 1,-4097,531442,-16773121,244140626,-2177317874,13841287202,
%T A321809 -68702695425,282430067923,-1000244144722,3138428376722,
%U A321809 -8913940970482,23298085122482,-56707753666594,129746582562692,-281406240452609,582622237229762
%N A321809 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^12.
%H A321809 Seiichi Manyama, <a href="/A321809/b321809.txt">Table of n, a(n) for n = 1..10000</a>
%H A321809 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 A321809 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A321809 G.f.: Sum_{k>=1} (-1)^(k+1)*k^12*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Dec 22 2018
%F A321809 Multiplicative with a(2^e) = -(2047*2^(12*e+1) + 8191)/4095, and a(p^e) = (p^(12*e+12) - 1)/(p^12 - 1) for p > 2. - _Amiram Eldar_, Nov 22 2022
%t A321809 a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^12 &]; Array[a, 50] (* _Amiram Eldar_, Nov 22 2022 *)
%o A321809 (PARI) apply( A321809(n)=sumdiv(n, d, (-1)^(n\d-d)*d^12), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y A321809 Column k=12 of A322083.
%Y A321809 Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
%K A321809 sign,mult
%O A321809 1,2
%A A321809 _N. J. A. Sloane_, Nov 23 2018