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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

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

M. 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).

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. A004009

Sequence in context: A009255 A062451 A065959 * A017669 A277065 A001158

Adjacent sequences:  A226330 A226331 A226332 * A226334 A226335 A226336

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Jun 04 2013

STATUS

approved

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Last modified December 10 12:15 EST 2019. Contains 329895 sequences. (Running on oeis4.)