A111580 Expansion of eta(q)^2 * eta(q^2) * eta(q^10)^3 / eta(q^5)^2 in powers of q.

1, -2, -2, 4, 1, 4, -6, -8, 7, -2, 12, -8, -12, 12, -2, 16, -16, -14, 20, 4, 12, -24, -22, 16, 1, 24, -20, -24, 30, 4, 32, -32, -24, 32, -6, 28, -36, -40, 24, -8, 42, -24, -42, 48, 7, 44, -46, -32, 43, -2, 32, -48, -52, 40, 12, 48, -40, -60, 60, -8, 62, -64, -42, 64, -12, 48, -66, -64, 44, 12, 72, -56, -72, 72

Offset

1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

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

Links

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

Shaun Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 318, Th. 4.1.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

Expansion of q * psi(q)^3 * psi(q^5) - 5*q^2 * psi(q) * psi(q^5)^3 in powers of q where psi() is a Ramanujan theta function.

Euler transform of period 10 sequence [-2, -3, -2, -3, 0, -3, -2, -3, -2, -4, ...].

G.f.: Sum_{k>0} Kronecker(k, 5) * k * x^k / (1 - x^(2*k)) = x * Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k)) * (1 + x^(5*k))^2 * (1 - x^(10*k)).

a(2*n) = -2*a(n).

From Amiram Eldar, Jan 28 2024: (Start)

a(n) = (-1)^(n+1) * A138558(n).

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . (End)

Example

G.f. = q - 2*q^2 - 2*q^3 + 4*q^4 + q^5 + 4*q^6 - 6*q^7 - 8*q^8 + 7*q^9 - 2*q^10 + ...

Mathematica

a[n_] := Sum[Mod[n/d, 2]*d*KroneckerSymbol[d, 5], {d, Divisors[n]}]; Table[a[n], {n, 1, 74}](* Jean-François Alcover, May 11 2012, after PARI *)
a[ n_] := SeriesCoefficient[ (1/16) (EllipticTheta[ 2, 0, q]^3 EllipticTheta[ 2, 0, q^5] - 5 EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^5]^3), {q, 0, 2 n}]; (* Michael Somos, Jul 12 2012 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d * kronecker( d, 5)))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^10 + A)^3 / eta(x^5 + A)^2, n))};
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^2)*eta(q^10)^3/eta(q^5)^2) \\ Altug Alkan, Apr 18 2018

Crossrefs

Cf. A138558.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A352707 A140819 A138558 * A066202 A027420 A116588

Adjacent sequences: A111577 A111578 A111579 * A111581 A111582 A111583

Keyword

sign,mult

Author

Michael Somos, Aug 08 2005

Status

approved