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A226333 Expansion of (E_4(q) - E_4(q^5)) / 240 in powers of q where E_4 is an Eisenstein series. 1
1, 9, 28, 73, 125, 252, 344, 585, 757, 1125, 1332, 2044, 2198, 3096, 3500, 4681, 4914, 6813, 6860, 9125, 9632, 11988, 12168, 16380, 15625, 19782, 20440, 25112, 24390, 31500, 29792, 37449, 37296, 44226, 43000, 55261, 50654, 61740, 61544, 73125, 68922, 86688 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
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.
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).
G.f.: Sum_{k>0} k^3 * x^k * (1 - x^(4*k)) / ((1 - x^k) * (1 - x^(5*k))).
a(n) = A004009(n) if n is not divisible by 5, else a(n) = 5^3 * a(n/5).
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/5^s) * zeta(s-3) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^4, where c = 26*Pi^4/9375 = 0.270147... . (End)
EXAMPLE
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 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, DivisorSigma[ 3, n] - If[ Divisible[ n, 5], DivisorSigma[ 3, n/5], 0]]
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[q^5])^4 + 13 q^2 ( QPochhammer[q^5]^5 / QPochhammer[ q])^2, {q, 0, n}]
PROG
(PARI) {a(n) = if( n<1, 0, sigma( n, 3) - if( n%5, 0, sigma( n/5, 3)))}
(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))}
(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)))))}
CROSSREFS
Sequence in context: A009255 A062451 A065959 * A017669 A277065 A001158
KEYWORD
nonn,easy,mult
AUTHOR
Michael Somos, Jun 04 2013
STATUS
approved

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Last modified June 14 05:17 EDT 2024. Contains 373393 sequences. (Running on oeis4.)