A113185 Expansion of (5*phi(q)*phi^3(q^5) - phi^3(q)*phi(q^5))/4 in powers of q where phi(q) is a Ramanujan theta function.

1, 1, -3, -2, 1, 1, 6, -6, -7, 7, -3, 12, -2, -12, 18, -2, 9, -16, -21, 20, 1, 12, -36, -22, 14, 1, 36, -20, -6, 30, 6, 32, -23, -24, 48, -6, 7, -36, -60, 24, -7, 42, -36, -42, 12, 7, 66, -46, -18, 43, -3, 32, -12, -52, 60, 12, 42, -40, -90, 60, -2, 62, -96, -42, 41, -12, 72, -66, -16, 44, 18, 72, -49, -72, 108, -2, 20

Offset

0,3

Comments

Ramanujan theta functions: f(q) := Prod_{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..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

References

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 249, Entry 8(ii).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of (eta(q^2)^5 * eta(q^10)^7)/(eta(q)*eta(q^4)*eta(q^5)^3 * eta(q^20)^3) in powers of q.

Euler transform of period 20 sequence [1, -4, 1, -3, 4, -4, 1, -3, 1, -8, 1, -3, 1, -4, 4, -3, 1, -4, 1, -4, ...].

a(n) is multiplicative with a(5^e) = 1, a(2^e) = ((-2)^(e+1)-1)/(-2-1)-2 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 9 (mod 10), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 7 (mod 10).

G.f.: (5phi(q) phi(q^5)^3 - phi(q)^3 phi(q^5))/4 where phi(q)=1+2(q+q^4+q^9+...).

a(n) = (-1)^n * A132069(n).

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . - Amiram Eldar, Jan 28 2024

Example

1 + q - 3*q^2 - 2*q^3 + q^4 + q^5 + 6*q^6 - 6*q^7 - 7*q^8 + 7*q^9 + ...

Mathematica

a[n_]:= SeriesCoefficient[(5*EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^5]^3 - EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, q^5])/4, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 16 2017 *)

Prog

(PARI) a(n)=if(n<1, n==0, (-1)^n*sumdiv(n, d, kronecker(5, d)*d*(-1)^d))
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^10+A)^7/(eta(x+A)*eta(x^4+A)*eta(x^5+A)^3*eta(x^20+A)^3), n))}
(PARI) {a(n)=local(A, p, e, a1); if(n<1, n==0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==5, 1, if(p>2, p*=kronecker(5, p); (p^(e+1)-1)/(p-1), a1=-3; for(i=2, e, a1=-2*a1-5); a1)))))}

Crossrefs

Cf. A132069.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A176669 A327615 A058280 * A132069 A259786 A254410

Adjacent sequences: A113182 A113183 A113184 * A113186 A113187 A113188

Keyword

sign,mult

Author

Michael Somos, Oct 17 2005, Mar 20 2008

Status

approved