cp's OEIS Frontend

A138951 Expansion of eta(q^2)^7 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^3 * eta(q^6)^3) in powers of q.

%I A138951 #17 Feb 16 2025 08:33:08
%S A138951 1,2,-2,-6,-2,4,6,0,-2,2,-4,0,6,4,0,-12,-2,4,-2,0,-4,0,0,0,6,6,-4,-6,
%T A138951 0,4,12,0,-2,0,-4,0,-2,4,0,-12,-4,4,0,0,0,4,0,0,6,2,-6,-12,-4,4,6,0,0,
%U A138951 0,-4,0,12,4,0,0,-2,8,0,0,-4,0,0,0,-2,4,-4,-18,0
%N A138951 Expansion of eta(q^2)^7 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^3 * eta(q^6)^3) in powers of q.
%C A138951 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)
%H A138951 G. C. Greubel, <a href="/A138951/b138951.txt">Table of n, a(n) for n = 0..10000</a>
%H A138951 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138951 L.-C. Shen, <a href="https://doi.org/10.1090/S0002-9939-1994-1212287-3">On the Modular Equations of Degree 3</a>, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1108, Eq. (3.24).
%H A138951 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138951 Expansion of (3 * phi(-q^3)^2 - phi(-q)^2) / 2 in powers of q where phi() is a Ramanujan theta function.
%F A138951 Expansion of phi(q) * phi(-q^2) * chi(-q^3) / chi(q^3) in powers of q where phi(), chi() are Ramanujan theta functions.
%F A138951 Euler transform of period 12 sequence [ 2, -5, 0, -2, 2, -4, 2, -2, 0, -5, 2, -2, ...].
%F A138951 Moebius transform is period 24 sequence [ 2, -4, -8, 0, 2, 16, -2, 0, 8, -4, -2, 0, 2, 4, -8, 0, 2, -16, -2, 0, 8, 4, -2, 0, ...].
%F A138951 a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = -1 if e>0, b(3^e) = -1 + 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 A138951 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129447.
%F A138951 a(12*n + 7) = a(12*n + 11) = 0.
%F A138951 G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 - x^(2*k) + x^(4*k)) / ((1 + x^(2*k))^2 * (1 - x^k + x^(2*k))^2).
%F A138951 a(n) = (-1)^n * A138949(n).
%e A138951 G.f. = 1 + 2*q - 2*q^2 - 6*q^3 - 2*q^4 + 4*q^5 + 6*q^6 - 2*q^8 + 2*q^9 - 4*q^10 + ...
%t A138951 a[ n_] := If[ n < 1, Boole[n == 0], -2 (-1)^n DivisorSum[ n, KroneckerSymbol[ -4, n/#] {1, 1, -2}[[Mod[#, 3, 1]]] &]]; (* _Michael Somos_, Sep 07 2015 *)
%t A138951 a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 4, 0, q^3]^2 - EllipticTheta[ 4, 0, q]^2) / 2, {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t A138951 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2] QPochhammer[ q^3] / QPochhammer[ -q^3], {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t A138951 a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ (Which[ # == 1, 1, # == 2, -1, # == 3, -1 + 2 (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* _Michael Somos_, Sep 07 2015 *)
%o A138951 (PARI) {a(n) = if( n<1, n==0, -2 * (-1)^n * sumdiv(n, d, kronecker(-4, n/d) * [ -2, 1, 1][d%3 + 1]))};
%o A138951 (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -1, p==3, -1 + 2 * (-1)^e, if(p%12 < 6, e+1, 1-e%2) )))};
%o A138951 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^6 + A)^3), n))};
%Y A138951 Cf. A138949.
%K A138951 sign
%O A138951 0,2
%A A138951 _Michael Somos_, Apr 03 2008