A256282 - OEIS

%I #21 Feb 16 2025 08:33:25

%S 1,-1,1,0,1,-2,0,0,1,-4,2,0,0,-2,0,0,1,-2,4,0,2,0,0,0,0,-3,2,0,0,-2,0,

%T 0,1,0,2,0,4,-2,0,0,2,-2,0,0,0,-8,0,0,0,-1,3,0,2,-2,0,0,0,0,2,0,0,-2,

%U 0,0,1,-4,0,0,2,0,0,0,4,-2,2,0,0,0,0,0,2

%N Expansion of f(-q^3) * psi(q^3)^3 / (psi(q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

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

%H G. C. Greubel, <a href="/A256282/b256282.txt">Table of n, a(n) for n = 0..10000</a>

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

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

%F Expansion of eta(q) * eta(q^6)^6 * eta(q^9) / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)^2) in powers of q.

%F Euler transform of period 18 sequence [ -1, 1, 1, 1, -1, -3, -1, 1, 0, 1, -1, -3, -1, 1, 1, 1, -1, -2, ...].

%F Moebius transform is a period 72 sequence.

%F a(n) = (-1)^n * A258256(n). a(2*n) = A258256(n). a(3*n + 1) = - A258277(n). a(3*n + 2) = A258278(n). a(4*n + 3) = 0.

%F a(n) = -b(n) where b() is multiplicative with a(0) = 1,  b(2^e) = -1 if e>0, b(3^e) = 2 * (1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - _Michael Somos_, Jun 06 2015

%e G.f. = 1 - q + q^2 + q^4 - 2*q^5 + q^8 - 4*q^9 + 2*q^10 - 2*q^13 + q^16 + ...

%t a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3 / (2 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}];

%t a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[#, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + #) &]]; (* _Michael Somos_, Jun 06 2015 *)

%t a[ n_] := If[ n < 1, Boole[ n==0 ], -Times @@ (Which[ # == 1, 1, # == 2, -1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])]; (* _Michael Somos_, Jun 06 2015 *)

%o (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + d)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^6 * eta(x^9 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^18 + A)^2), n))};

%Y Cf. A258256, A258277, A258278.

%K sign,changed

%O 0,6

%A _Michael Somos_, Jun 02 2015