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%I #34 Dec 12 2023 08:00:43
%S 1,28,126,344,757,1332,2198,3528,4914,6860,9632,12168,15751,20440,
%T 24390,29792,37296,43344,50654,61544,68922,79508,95382,103824,117993,
%U 137592,148878,167832,192080,205380,226982,260408,276948,300764,340704,357912
%N a(n) = sigma_3(2*n+1).
%H Harvey P. Dale, <a href="/A045823/b045823.txt">Table of n, a(n) for n = 0..1000</a>
%F Expansion of q^(-1) * ( E_4(q) - 9 * E_4(q^2) + 8 * E_4(q^4) ) / 240 in powers of q^2. - _Michael Somos_, Nov 29 2007
%F Expansion of q^(-1) * (eta(q^2)^24 + eta(q)^16 * eta(q^4)^8) / (2 * eta(q)^8 * eta(q^2)^8) in powers of q^2. - _Michael Somos_, Nov 29 2007
%F a(n) = b(2*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1) if p>2. - _Michael Somos_, Nov 29 2007
%F G.f.: (theta_3(q)^8 - theta_4(q)^8) / (32*q) = Sum_{n>=0} sigma_3(2*n+1)*q^(2*n). - _Paul D. Hanna_, Jun 02 2018
%F Sum_{k=0..n} a(k) ~ (15*zeta(4)/8) * n^4. - _Amiram Eldar_, Dec 12 2023
%e q + 28*q^3 + 126*q^5 + 344*q^7 + 757*q^9 + 1332*q^11 + 2198*q^13 + ...
%p A045823 := proc(n)
%p numtheory[sigma][3](2*n+1) ;
%p end proc:
%p seq(A045823(n),n=0..30) ; # _R. J. Mathar_, Nov 25 2018
%t DivisorSigma[3, Range[1, 75, 2]] (* _Harvey P. Dale_, Jan 11 2015 *)
%o (PARI) {a(n) = if( n<0, 0, sigma(2 * n + 1, 3))} /* _Michael Somos_, Nov 29 2007 */
%o (PARI) {a(n) = local(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( (eta(x^2 + A)^24 + eta(x + A)^16 * eta(x^4 + A)^8) / (2 * eta(x + A)^8 * eta(x^2 + A)^8), n))} /* _Michael Somos_, Nov 29 2007 */
%o (Magma) [DivisorSigma(3, 2*n+1): n in [0..40]]; // _Vincenzo Librandi_, Jun 02 2019
%Y Equals A045819/2.
%Y Bisection of A001158.
%Y Cf. A008438, A013662, A091986.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Benoit Cloitre_, Apr 12 2003
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