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%I #19 Jan 22 2024 00:12:50
%S 1,9,28,73,125,252,344,585,757,1125,1332,2044,2198,3096,3500,4681,
%T 4914,6813,6860,9125,9632,11988,12168,16380,15625,19782,20440,25112,
%U 24390,31500,29792,37449,37296,44226,43000,55261,50654,61740,61544,73125,68922,86688
%N Expansion of (E_4(q) - E_4(q^5)) / 240 in powers of q where E_4 is an Eisenstein series.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A226333/b226333.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of q * (f(-q) * f(-q^5))^4 + 13 * q^2 * (f(-q^5)^5 / f(-q))^2 in powers of q where f() is a Ramanujan theta function.
%F a(n) is multiplicative with a(p^e) = p^(3*e) if p=5, else a(p^e) = (p^(3*(e+1)) - 1) / (p^3 - 1).
%F G.f.: Sum_{k>0} k^3 * x^k * (1 - x^(4*k)) / ((1 - x^k) * (1 - x^(5*k))).
%F a(n) = A004009(n) if n is not divisible by 5, else a(n) = 5^3 * a(n/5).
%F From _Amiram Eldar_, Sep 12 2023: (Start)
%F Dirichlet g.f.: (1 - 1/5^s) * zeta(s-3) * zeta(s).
%F Sum_{k=1..n} a(k) ~ c * n^4, where c = 26*Pi^4/9375 = 0.270147... . (End)
%e q + 9*q^2 + 28*q^3 + 73*q^4 + 125*q^5 + 252*q^6 + 344*q^7 + 585*q^8 + 757*q^9 + ...
%t a[ n_] := If[ n < 1, 0, DivisorSigma[ 3, n] - If[ Divisible[ n, 5], DivisorSigma[ 3, n/5], 0]]
%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[q^5])^4 + 13 q^2 ( QPochhammer[q^5]^5 / QPochhammer[ q])^2, {q, 0, n}]
%o (PARI) {a(n) = if( n<1, 0, sigma( n, 3) - if( n%5, 0, sigma( n/5, 3)))}
%o (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^5 + A))^4 + 13 * x * (eta(x^5 + A)^5 / eta(x + A))^2, n))}
%o (PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==5, p^(3*e), (p^(3*e+3) - 1) / (p^3 - 1)))))}
%Y Cf. A000122, A000700, A004009, A010054, A121373.
%K nonn,easy,mult
%O 1,2
%A _Michael Somos_, Jun 04 2013
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