A140727 Expansion of (phi(q) * phi(q^15) - phi(q^3) * phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, 0, -1, 1, -1, 0, 0, -2, 1, 0, 0, -1, 0, 0, 1, 3, -2, 0, 2, -1, 0, 0, -2, 2, 1, 0, -1, 0, 0, 0, 2, -4, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, -2, -3, 1, 0, 2, 0, -2, 0, 0, 0, -2, 0, 0, 1, 2, 0, 0, 5, 0, 0, 0, -2, 2, 0, 0, -2, 0, 0, -1, 2, 0, 0, 2, -3, 1, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, -2, -2, 0, -2, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset

1,8

Comments

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

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(v).

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * f(-q^2) * f(-q^30) * chi(q^3) * chi(q^5) in powers of q where f(), chi() are Ramanujan theta functions.

Expansion of eta(q^2) * eta(q^6)^2 * eta(q^10)^2 * eta(q^30) / (eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20)) in powers of q.

Euler transform of period 60 sequence [ 0, -1, 1, -1, 1, -2, 0, -1, 1, -2, 0, -1, 0, -1, 2, -1, 0, -2, 0, -1, 1, -1, 0, -1, 1, -1, 1, -1, 0, -4, 0, -1, 1, -1, 1, -1, 0, -1, 1, -1, 0, -2, 0, -1, 2, -1, 0, -1, 0, -2, 1, -1, 0, -2, 1, -1, 1, -1, 0, -2, ...].

a(n) is multiplicative with a(2^e) = (-1)^e * (e-1) if e>0. a(3^e) = a(5^e) = (-1)^e, a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (-1)^e * (e+1) if p == 2, 8 (mod 15), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15).

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = 60^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

a(n) = -(-1)^n * A140728(n). abs(a(n)) = A122855(n).

Example

G.f. = q - q^3 + q^4 - q^5 - 2*q^8 + q^9 - q^12 + q^15 + 3*q^16 - 2*q^17 + ...

Mathematica

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ 5, #] KroneckerSymbol[ -3, n/#] &]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2] QPochhammer[ q^30] QPochhammer[ -q^3, q^6] QPochhammer[ -q^5, q^10], {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (-1)^(n + d) * kronecker(5, d) * kronecker(-3, n/d)))}
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-1)^e * (e-1), p==3 || p==5, (-1)^e, kronecker(p, 15)==1, (e+1) * (-1)^(e*valuation(p%15, 2)), (1 + (-1)^e) / 2)))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^2 * eta(x^10 + A)^2 * eta(x^30 + A) / (eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A)), n))};

Crossrefs

Cf. A122855, A140728.

Sequence in context: A359815 A260649 A122855 * A140728 A254110 A331289

Adjacent sequences: A140724 A140725 A140726 * A140728 A140729 A140730

Keyword

sign,mult

Author

Michael Somos, May 29 2008

Status

approved