OFFSET
1,2
COMMENTS
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
From Michael Somos, Jan 17 2015: (Start)
Expansion of b(q) * (b(q^4) - b(q)) / (3*b(q^2)) in powers of q where b() is a cubic AGM theta function.
Expansion of q * chi(-q)^3 * phi(-q^2) * psi(q^3) / chi(-q^6)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q * phi(-q)^2 * psi(q^6)^2 / (psi(-q) * psi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of q * f(q) * f(-q, -q^5)^4 / f(q^3)^3 in powers of q where f() is a Ramanujan theta function. (End)
Expansion of (eta(q) * eta(q^12))^3 / (eta(q^2) * eta(q^3) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -3, -2, -2, -1, -3, 0, -3, -1, -2, -2, -3, -2, ...].
Moebius transform is period 12 sequence [ 1, -4, 0, 6, -1, 0, 1, -6, 0, 4, -1, 0, ...].
Multiplicative with a(2^e) = 3(-1)^e if e>0, a(3^e)=1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 2 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(-3, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
a(n) = -(-1)^n * A244375(n). a(6*n + 5) = 0, a(3*n) = a(n).
a(2*n) = -3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n + 1) = A129576(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 2) = -3 * A033687(n).
Sum_{k=1..n} abs(a(k)) ~ (Pi/sqrt(3)) * n. - Amiram Eldar, Jan 23 2024
EXAMPLE
G.f. = q - 3*q^2 + q^3 + 3*q^4 - 3*q^6 + 2*q^7 - 3*q^8 + q^9 + 3*q^12 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(3/2)] / (2 q^(3/8)), {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, JacobiSymbol[ -3, n/#] {1, -2, 1, 0}[[Mod[#, 4, 1]]] &]]; (* Michael Somos, Jan 17 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-3, n/d)*[0, 1, -2, 1][d%4 + 1]))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^12 + A))^3/ (eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^6 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 106); A[2] - 3*A[3] + A[4] + 3*A[5]; /* Michael Somos, Jan 17 2015 */
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Sep 02 2005
STATUS
approved