OFFSET
0,2
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
FORMULA
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
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
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
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
Sum_{k=0..n} a(k) ~ (15*zeta(4)/8) * n^4. - Amiram Eldar, Dec 12 2023
EXAMPLE
q + 28*q^3 + 126*q^5 + 344*q^7 + 757*q^9 + 1332*q^11 + 2198*q^13 + ...
MAPLE
A045823 := proc(n)
numtheory[sigma][3](2*n+1) ;
end proc:
seq(A045823(n), n=0..30) ; # R. J. Mathar, Nov 25 2018
MATHEMATICA
DivisorSigma[3, Range[1, 75, 2]] (* Harvey P. Dale, Jan 11 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, sigma(2 * n + 1, 3))} /* Michael Somos, Nov 29 2007 */
(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 */
(Magma) [DivisorSigma(3, 2*n+1): n in [0..40]]; // Vincenzo Librandi, Jun 02 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Benoit Cloitre, Apr 12 2003
STATUS
approved