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
Expansion of (b(q) - b(q^4)) * (b(q) - 2*b(q^4)) / (3* b(q^2)) = b(q^2)^2 * (b(q^4) - b(q)) / (3 * b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function.
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 * 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. - Michael Somos, Jan 17 2015
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. - Michael Somos, Jan 17 2015
Expansion of eta(q^2)^8 * eta(q^3) * eta(q^12)^4 / (eta(q)^3 * eta(q^4)^4 * eta(q^6)^4) in powers of q. - Michael Somos, Jan 17 2015
a(n) is multiplicative with a(2^e) = 3 * (-1)^(e+1) 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 == 5 (mod 6).
Euler transform of period 12 sequence [ 3, -5, 2, -1, 3, -2, 3, -1, 2, -5, 3, -2, ...].
Moebius transform is period 12 sequence [ 1, 2, 0, -6, -1, 0, 1, 6, 0, -2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A244339.
a(2*n) = 3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0.
a(n) = -(-1)^n * A112298(n). - Michael Somos, Jan 17 2015
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_] := If[ n < 1, 0, Sum[ Mod[ n/d, 2] {1, 3, 0, -3, -1, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^8 QPochhammer[ q^3] QPochhammer[ q^12]^4 / (QPochhammer[ q]^3 QPochhammer[ q^4]^4 QPochhammer[ q^6]^4), {q, 0, n}];
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)] / (2^(1/2) q^(3/8)), {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, (n/d%2) * [0, 1, 3, 0, -3, -1][d%6 + 1]))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)^4 / (eta(x + A)^3 * eta(x^4 + A)^4 * eta(x^6 + A)^4), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 + x^k + x^(2*k)) * [0, 1, 4, 1][k%4 + 1], x * O(x^n)), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 - x^(2*k)) * [0, 1, 3, 0, -3, -1][k%6 + 1], x * O(x^n)), n))};
(PARI) {a(n) = my(A); if( n<1, 0, A = factor(n); prod( j=1, matsize(A)[1], if( p = A[j, 1], e = A[j, 2]; if( p==2, 3 * (-1)^(e+1), if( p==3, 1, if( p%6 == 1, e+1, (1 + (-1)^e) / 2))))))};
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 82); A[2] + 3*A[3] + A[4] - 3*A[5]; /* Michael Somos, Jan 17 2015 */
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Jun 26 2014
STATUS
approved