A226333 Expansion of (E_4(q) - E_4(q^5)) / 240 in powers of q where E_4 is an Eisenstein series.

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

Offset

1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions.

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

Cf. A000122, A000700, A004009, A010054, A121373.

Sequence in context: A009255 A062451 A065959 * A017669 A277065 A001158

Adjacent sequences: A226330 A226331 A226332 * A226334 A226335 A226336

Keyword

nonn,easy,mult

Author

Michael Somos, Jun 04 2013

Status

approved