OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael D. Hirschhorn and James A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.6.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of (eta(q)eta(q^8))^4/(eta(q^2)eta(q^4))^2 in powers of q.
Euler transform of period 8 sequence [ -4, -2, -4, 0, -4, -2, -4, -4, ...].
Multiplicative with a(2)=-4, a(2^e)=0 if e>1, a(p^e)=(p^(e+1)-1)/(p-1) if p>2.
a(4n)=0. a(4n+2)=-4*sigma(2n+1). a(2n+1)=sigma(2n+1).
G.f. is Fourier series of a weight 2 level 8 cusp form. f(-1/ (8 t)) = -8 t^2 f(t) where q = exp(2 Pi i t).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w)= ( v* (v+2*w)* (u+2*v))^2 -16* (u*w)^3 -u*v*w* (u +2*v +4*w) *(u^2 +16*v^2 +16*w^2 +10*u*v +28*u*w +40*v*w).
Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * (1 - 1/2^s) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023
EXAMPLE
q - 4*q^2 + 4*q^3 + 6*q^5 - 16*q^6 + 8*q^7 + 13*q^9 - 24*q^10 + 12*q^11 + ...
MATHEMATICA
a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, q^2] *EllipticTheta[3, 0, -q])^2/4, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)
PROG
(PARI) {a(n)=if(n<1, 0, if(n%2, sigma(n), if(n/2%2, -4*sigma(n/2), 0)))}
(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^8+A))^4/(eta(x^2+A)*eta(x^4+A))^2, n))}
CROSSREFS
KEYWORD
sign,easy,mult
AUTHOR
Michael Somos, Jul 30 2006, May 28 2007
STATUS
approved