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%I #19 Nov 22 2022 22:00:50
%S 1,-513,19684,-261633,1953126,-10097892,40353608,-133955073,387440173,
%T -1001953638,2357947692,-5149983972,10604499374,-20701400904,
%U 38445332184,-68584996353,118587876498,-198756808749,322687697780,-511002214758
%N a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^9.
%H Seiichi Manyama, <a href="/A321565/b321565.txt">Table of n, a(n) for n = 1..10000</a>
%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&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 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F G.f.: Sum_{k>=1} (-1)^(k+1)*k^9*x^k/(1 + x^k). - _Ilya Gutkovskiy_, Dec 22 2018
%F Multiplicative with a(2^e) = -3*(85*2^(9*e+1) + 341)/511, and a(p^e) = (p^(9*e+9) - 1)/(p^9 - 1) for p > 2. - _Amiram Eldar_, Nov 22 2022
%t CoefficientList[Series[Sum[(-1)^(k+1) k^9 x^k/(1+x^k),{k,20}],{x,0,20}],x] (* _Harvey P. Dale_, Apr 09 2019 *)
%t a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^9 &]; Array[a, 25] (* _Amiram Eldar_, Nov 22 2022 *)
%o (PARI) apply( A321565(n)=sumdiv(n, d, (-1)^(n\d-d)*d^9), [1..30]) \\ _M. F. Hasler_, Nov 26 2018
%Y Column k=9 of A322083.
%Y Cf. A321543 - A321564, A321807 - A321836 for similar sequences.
%K sign,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 23 2018
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