A107760 - OEIS

%I #36 Dec 28 2023 01:51:45

%S 1,3,3,3,3,0,3,6,3,3,0,0,3,6,6,0,3,0,3,6,0,6,0,0,3,3,6,3,6,0,0,6,3,0,

%T 0,0,3,6,6,6,0,0,6,6,0,0,0,0,3,9,3,0,6,0,3,0,6,6,0,0,0,6,6,6,3,0,0,6,

%U 0,0,0,0,3,6,6,3,6,0,6,6,0,3,0,0,6,0,6,0,0,0,0,12,0,6,0,0,3,6,9,0,3,0,0,6,6

%N Expansion of eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2) in powers of q.

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

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.42).

%H G. C. Greubel, <a href="/A107760/b107760.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u^2*w + 4 * v*w^2 - 4 * v^2*w - 2 * u*v*w.

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 - u2) * (u1 - u2 - u3 + u6) - 3 * u6 * (u2 - u6).

%F Expansion of psi(q)^3 / psi(q^3) in powers of q where psi() is a Ramanujan theta function.

%F Expansion of (a(q) + a(q^2)) / 2 = b(q^2)^2 / b(q) in powers of q where a(), b() are cubic AGM theta functions. - _Michael Somos_, Aug 30 2008

%F Euler transform of period 6 sequence [ 3, -3, 2, -3, 3, -2, ...].

%F Moebius transform is period 6 sequence [ 3, 0, 0, 0, -3, 0, ...]. - _Michael Somos_, Aug 11 2009

%F a(n) = 3 * b(n) unless n=0 and b() is multiplicative with b(p^e) = 1 if p=2 or p=3; b(p^e) = 1+e if p == 1 (mod 6); b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6). - _Michael Somos_, Aug 11 2009

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123330. - _Michael Somos_, Aug 11 2009

%F G.f.: (Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1)))^3 / (Product_{k>0} (1 - x^(6*k)) / (1 - x^(6*k - 3))). - _Michael Somos_, Aug 11 2009

%F a(n) = 3 * A035178(n) unless n=0. a(n) = (-1)^n * A132973. a(2*n) = a(3*n) = a(n). a(6*n + 5) = 0. a(2*n + 1) = 3 * A033762. a(3*n + 1) = 3 * A033687(n). a(4*n + 1) = 3 * A112604(n). a(4*n + 3) = 3 * A112605(n). a(6*n + 1) = 3 * A097195(n). Convolution inverse of A132979.

%F a(8*n + 1) = 3 * A112606(n). a(8*n + 3) = 3* A112608(n). a(8*n + 5) = 6 * A112607(n-1). a(8*n + 7) = 6 * A112609(n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3)/2 = 2.720699... . - _Amiram Eldar_, Dec 28 2023

%e G.f. = 1 + 3*q + 3*q^2 + 3*q^3 + 3*q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + ...

%t a[ n_] := If[ n < 1, Boole[n == 0], 3 Times @@ (Which[ # < 5, 1, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1 ] & @@@ FactorInteger@n)]; (* _Michael Somos_, Aug 04 2015 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^3 / (4 EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* _Michael Somos_, Aug 04 2015 *)

%t QP = QPochhammer; s = QP[q^3]*(QP[q^2]^6/(QP[q]^3*QP[q^6]^2)) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 24 2015 *)

%o (PARI) {a(n) = if( n<1, n==0, 3 * direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -12, p) * X)))[n])};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^2 + A)^6 / (eta(x^6 + A)^2 * eta(x + A)^3), n))};

%o (PARI) {a(n) = if ( n<1, n==0, 3 * sumdiv( n, d, kronecker( -12, d)))};

%o (Sage) A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 3*A[1] # _Michael Somos_, Sep 27 2013

%o (Magma) A := Basis( ModularForms( Gamma1(6), 1), 88); A[1] + 3*A[2]; /* _Michael Somos_, Aug 04 2015 */

%Y Cf. A033687, A033762, A035178, A097195, A112604, A112605, A123330, A132973, A132979.

%Y Cf. A112606, A112607, A112608, A112609. - _Michael Somos_, Sep 27 2013

%Y Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

%K nonn,easy

%O 0,2

%A _Michael Somos_, May 24 2005