A121455 Expansion of q*(phi(-q)psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

1, -4, 4, 0, 6, -16, 8, 0, 13, -24, 12, 0, 14, -32, 24, 0, 18, -52, 20, 0, 32, -48, 24, 0, 31, -56, 40, 0, 30, -96, 32, 0, 48, -72, 48, 0, 38, -80, 56, 0, 42, -128, 44, 0, 78, -96, 48, 0, 57, -124, 72, 0, 54, -160, 72, 0, 80, -120, 60, 0, 62, -128, 104, 0, 84, -192, 68, 0, 96, -192, 72, 0, 74, -152

Offset: 1

Author

Michael Somos, Jul 30 2006, May 28 2007

Keywords

• sign
• easy
• mult

Comments

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k>=0} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			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 + ...
		

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.

Crossrefs

Cf. A000122, A000700, A010054, A121373.

Programs

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 *)

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))}

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