A113186 Expansion of (25phi(q)phi^3(q^5)-phi^5(q)/phi(q^5)-24)/40 in powers of q where phi(q) is a Ramanujan theta function.

1, -1, -2, -1, 1, 2, -6, -1, 7, -1, 12, 2, -12, 6, -2, -1, -16, -7, 20, -1, 12, -12, -22, 2, 1, 12, -20, 6, 30, 2, 32, -1, -24, 16, -6, -7, -36, -20, 24, -1, 42, -12, -42, -12, 7, 22, -46, 2, 43, -1, 32, 12, -52, 20, 12, 6, -40, -30, 60, 2, 62, -32, -42, -1, -12, 24, -66, 16, 44, 6, 72, -7, -72, 36, -2, -20, -72, -24, 80, -1

Offset

1,3

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(iii).

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

a(n) is multiplicative with a(2^e) = -1 if e>0, a(5^e) = 1, 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.: (25phi(q) phi(q^5)^3 - phi(q)^5/phi(q^5)-24)/40 where phi(q) = 1 + 2(q+q^4+q^9+...).

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(18*sqrt(5)) = 0.245212... . - Amiram Eldar, Jan 28 2024

Mathematica

a[n_]:= SeriesCoefficient[(25*EllipticTheta[3, 0, q]*(EllipticTheta[3, 0, q^5])^3 - (EllipticTheta[3, 0, q])^5/EllipticTheta[3, 0, q^5] - 24)/40, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Mar 07 2018 *)

Prog

(PARI) a(n)=if(n<1, 0, (-1)^n*sumdiv(n, d, kronecker(20, d)*d*(-1)^d))
(PARI) {a(n)=local(A, p, e); 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==2, -1, if(p==5, 1, p*=kronecker(5, p); (p^(e+1)-1)/(p-1))))))}

Crossrefs

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A156588 A278543 A374571 * A206497 A123218 A298608

Adjacent sequences: A113183 A113184 A113185 * A113187 A113188 A113189

Keyword

sign,mult

Author

Michael Somos, Oct 17 2005

Status

approved