cp's OEIS Frontend

A138745 Expansion of eta(q) * eta(q^3) * eta(q^4)^3 / (eta(q^2)^2 * eta(q^12)) in powers of q.

%I A138745 #22 Feb 16 2025 08:33:08
%S A138745 1,-1,1,-3,1,-2,3,0,1,-1,2,0,3,-2,0,-6,1,-2,1,0,2,0,0,0,3,-3,2,-3,0,
%T A138745 -2,6,0,1,0,2,0,1,-2,0,-6,2,-2,0,0,0,-2,0,0,3,-1,3,-6,2,-2,3,0,0,0,2,
%U A138745 0,6,-2,0,0,1,-4,0,0,2,0,0,0,1,-2,2,-9,0,0,6,0
%N A138745 Expansion of eta(q) * eta(q^3) * eta(q^4)^3 / (eta(q^2)^2 * eta(q^12)) in powers of q.
%C A138745 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138745 G. C. Greubel, <a href="/A138745/b138745.txt">Table of n, a(n) for n = 0..10000</a>
%H A138745 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138745 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138745 Expansion of (theta_4(q)^2 + 3 * theta_4(q^3)^2) / 4 in powers of q.
%F A138745 Expansion of psi(-q) * psi(q^2) * chi(-q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.
%F A138745 Euler transform of period 12 sequence [ -1, 1, -2, -2, -1, 0, -1, -2, -2, 1, -1, -2, ...].
%F A138745 Moebius transform is period 24 sequence [ -1, 2, -2, 0, -1, 4, 1, 0, 2, 2, 1, 0, -1, -2, -2, 0, -1, -4, 1, 0, 2, -2, 1, 0, ...].
%F A138745 a(n) = -b(n) where b() is multiplicative with b(2^e) = -1 if e>0, b(3^e) = 2 - (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
%F A138745 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A125079.
%F A138745 G.f.: 1 + Sum_{k>0} (-1)^k * ( f(6*k - 1) + 2 * f(6*k - 3) + f(6*k - 5) ) where f(k) := x^k / (1 + x^k).
%F A138745 a(12*n + 7) = a(12*n + 11) = 0.
%F A138745 a(n) = - A138746(n) unless n=0. a(n) = (-1)^n * A125061(n).
%F A138745 a(2*n) = A125061(n). a(2*n + 1) = - A138741(n).
%e A138745 G.f. = 1 - q + q^2 - 3*q^3 + q^4 - 2*q^5 + 3*q^6 + q^8 - q^9 + 2*q^10 + ...
%t A138745 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q]^2 + 3 EllipticTheta[ 4, 0, q^3]^2) / 4, {q, 0, n}]; (* _Michael Somos_, Sep 08 2015 *)
%t A138745 a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* _Michael Somos_, Sep 08 2015 *)
%t A138745 a[ n_] := If[ n < 1, Boole[n == 0], - Times @@ (Which[ # < 3, -(-1)^#, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* _Michael Somos_, Sep 08 2015 *)
%t A138745 QP = QPochhammer; s = QP[q]*QP[q^3]*(QP[q^4]^3/(QP[q^2]^2*QP[q^12])) + O[q]^100; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%o A138745 (PARI) {a(n) = if( n<1, n==0, (-1)^n * sumdiv(n, d, ((d%2) * ((d%3==0) + 1)) * (-1)^(d\6)))};
%o A138745 (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); - prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2 )))};
%o A138745 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^3 / (eta(x^2 + A)^2 * eta(x^12 + A)), n))};
%Y A138745 Cf. A125061, A125079, A138741, A138746.
%K A138745 sign
%O A138745 0,4
%A A138745 _Michael Somos_, Mar 27 2008