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A096961 a(n) = Sum_{0

%I A096961 #31 Jan 09 2023 01:46:07
%S A096961 1,128,2188,16384,78126,280064,823544,2097152,4785157,10000128,
%T A096961 19487172,35848192,62748518,105413632,170939688,268435456,410338674,
%U A096961 612500096,893871740,1280016384,1801914272,2494358016,3404825448
%N A096961 a(n) = Sum_{0<d|n, n/d odd} d^7.
%H A096961 Seiichi Manyama, <a href="/A096961/b096961.txt">Table of n, a(n) for n = 1..10000</a>
%H A096961 H. H. Chan and C. Krattenthaler, <a href="https://arxiv.org/abs/math/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.
%H A096961 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 A096961 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F A096961 G.f.: Sum_{k>0} k^7 * x^k / (1 - x^(2*k)).
%F A096961 Expansion of (E_8(q) - E_8(q^2)) / 480 in powers of q where E_8() is an Eisenstein series (A008410). - _Michael Somos_, Jan 09 2015
%F A096961 From _Amiram Eldar_, Nov 02 2022: (Start)
%F A096961 Multiplicative with a(2^e) = 2^(7*e) and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
%F A096961 Sum_{k=1..n} a(k) ~ c * n^8, where c = 255*zeta(8)/2048 = 17*Pi^8/1290240 = 0.125019... . (End)
%F A096961 Dirichlet g.f.: zeta(s)*zeta(s-7)*(1-1/2^s). - _Amiram Eldar_, Jan 09 2023
%e A096961 G.f. = q + 128*q^2 + 2188*q^3 + 16384*q^4 + 78126*q^5 + 280064*q^6 + 823544*q^7 + ...
%t A096961 a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^8, u4 = QPochhammer[ q^4]^8}, q u2 (u1^2 + 136 q u4 u1 + 2176 q^2 u4^2 ) / u1], {q, 0, n}]; (* _Michael Somos_, Jun 04 2013 *)
%t A096961 a[ n_] := If[ n < 1, 0, Sum[ d^7 Mod[ n/d, 2], {d, Divisors[ n]}]]; (* _Michael Somos_, Jan 09 2015 *)
%o A096961 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^7))};
%o A096961 (Sage) ModularForms( Gamma0(2), 8, prec=24).2; # _Michael Somos_, Jun 04 2013
%o A096961 (Magma) A := Basis( ModularForms( Gamma0(2), 8), 24); A[2] + 128*A[3]; /* _Michael Somos_, Nov 30 2014 */
%Y A096961 Cf. A007331, A008410, A013666, A096960, A096962, A096963.
%K A096961 nonn,mult
%O A096961 1,2
%A A096961 _Ralf Stephan_, Jul 18 2004