The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A244375 Expansion of (a(q) + 3*a(q^2) - 4*a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function. 4
 1, 3, 1, -3, 0, 3, 2, 3, 1, 0, 0, -3, 2, 6, 0, -3, 0, 3, 2, 0, 2, 0, 0, 3, 1, 6, 1, -6, 0, 0, 2, 3, 0, 0, 0, -3, 2, 6, 2, 0, 0, 6, 2, 0, 0, 0, 0, -3, 3, 3, 0, -6, 0, 3, 0, 6, 2, 0, 0, 0, 2, 6, 2, -3, 0, 0, 2, 0, 0, 0, 0, 3, 2, 6, 1, -6, 0, 6, 2, 0, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 Michael Somos, Introduction to Ramanujan theta functions 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 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 Cf. A033687, A033762, A093829, A112298, A122861, A244339. Sequence in context: A091422 A201659 A253625 * A253626 A112298 A011430 Adjacent sequences: A244372 A244373 A244374 * A244376 A244377 A244378 KEYWORD sign,mult AUTHOR Michael Somos, Jun 26 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 28 14:02 EDT 2023. Contains 361595 sequences. (Running on oeis4.)