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%I #19 Dec 28 2023 01:52:23
%S 1,3,2,0,1,0,2,6,2,0,0,0,3,3,2,0,0,0,2,6,2,0,2,0,1,6,2,0,0,0,2,0,4,0,
%T 0,0,2,9,0,0,1,0,4,6,2,0,0,0,2,0,2,0,0,0,2,6,2,0,2,0,1,6,4,0,0,0,0,6,
%U 2,0,0,0,4,3,2,0,2,0,2,6,0,0,0,0,3,0,2
%N Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A138741/b138741.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 Expansion of q^(-1/2) * (theta_2(q)^2 + 3 * theta_2(q^3)^2) / 4 in powers of q.
%F Expansion of phi(q) * psi(q) * psi(q^3) / phi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
%F Euler transform of period 12 sequence [ 3, -4, 2, -2, 3, -2, 3, -2, 2, -4, 3, -2, ...].
%F Moebius transform is period 24 sequence [ 1, -1, 2, 0, 1, -2, -1, 0, -2, -1, -1, 0, 1, 1, 2, 0, 1, 2, -1, 0, -2, 1, -1, 0, ...].
%F a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1 + (-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 G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132003.
%F a(6*n + 3) = a(6*n + 5) = 0.
%F a(n) = (-1)^n * A116604(n). a(2*n) = A008441(n).
%F a(6*n) = A002175(n). a(6*n + 1) = 3 * A008441(n). a(6*n + 2) = 2 * A121444(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - _Amiram Eldar_, Dec 28 2023
%e G.f. = 1 + 3*x + 2*x^2 + x^4 + 2*x^6 + 6*x^7 + 2*x^8 + 3*x^12 + 3*x^13 + ...
%e G.f. = q + 3*q^3 + 2*q^5 + q^9 + 2*q^13 + 6*q^15 + 2*q^17 + 3*q^25 + 3*q^27 + ...
%t a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* _Michael Somos_, Sep 08 2015 *)
%t a[ n_] := SeriesCoefficient[ x^(-1/2) (EllipticTheta[ 2, 0, x]^2 + 3 EllipticTheta[ 2, 0, x^3]^2) / 4, {x, 0, n}]; (* _Michael Somos_, Sep 08 2015 *)
%t a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, # == 3, 2 - (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger[2 n + 1])]; (* _Michael Somos_, Sep 08 2015 *)
%t QP = QPochhammer; s = QP[q^2]^7*QP[q^3]*QP[q^12]^2 / (QP[q]^3*QP[q^4]^2* QP[q^6]^3) + O[q]^90; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 24 2015 *)
%o (PARI) {a(n) = if( n<0, 0, sumdiv( 2*n + 1, d, (-1)^(d\6) * [0, 1, 0, 2, 0, 1][d%6 + 1]))};
%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2 )))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)^3), n))};
%Y Cf. A002175, A008441, A019669, A116604, A121444, A132003.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Mar 27 2008
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