A132069 Expansion of eta(q) * eta(q^2)^2 * eta(q^5)^3 / eta(q^10)^2 in powers of q.

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, 72

Offset

0,3

Comments

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

Denoted by z(q) = q d/dq log k(q) in Cooper (2009) where k() is the g.f. of A112274. - Michael Somos, Jul 08 2012

References

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 253 Eq. (8.12)

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Shaun Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 312, eq. (1.4).

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

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

Euler transform of period 10 sequence [-1, -3, -1, -3, -4, -3, -1, -3, -1, -4, ...].

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

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129303.

a(n) = (-1)^n * A113185(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

G.f. = 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_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ 5, #] # (-1)^# &]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^5]^3 / QPochhammer[ q^10]^2, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^5]^3 - EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^5])/4, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

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

Crossrefs

Cf. A112274, A113185, A129303.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A327615 A058280 A113185 * A259786 A254410 A073201

Adjacent sequences: A132066 A132067 A132068 * A132070 A132071 A132072

Keyword

sign

Author

Michael Somos, Aug 08 2007, Mar 20 2008

Status

approved