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%I A136748 #17 Jan 28 2024 02:00:53
%S A136748 1,-1,1,-3,0,-1,2,3,1,0,0,-3,2,-2,0,-3,0,-1,2,0,2,0,0,3,1,-2,1,-6,0,0,
%T A136748 2,3,0,0,0,-3,2,-2,2,0,0,-2,2,0,0,0,0,-3,3,-1,0,-6,0,-1,0,6,2,0,0,0,2,
%U A136748 -2,2,-3,0,0,2,0,0,0,0,3,2,-2,1,-6,0,-2,2,0
%N A136748 Expansion of (a(q) - a(q^2) - 4*a(q^4) + 4*a(q^8)) / 6 in powers of q where a() is a cubic AGM theta function.
%C A136748 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A136748 G. C. Greubel, <a href="/A136748/b136748.txt">Table of n, a(n) for n = 1..1000</a>
%H A136748 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A136748 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A136748 Expansion of eta(q) * eta(q^3) * eta(q^4)^4 * eta(q^24)^2 / (eta(q^2) * eta(q^8) * eta(q^12))^2 in powers of q.
%F A136748 Euler transform of period 24 sequence [ -1, 1, -2, -3, -1, 0, -1, -1, -2, 1, -1, -2, -1, 1, -2, -1, -1, 0, -1, -3, -2, 1, -1, -2, ...].
%F A136748 a(n) is multiplicative with a(2) = -1, a(2^e) = -3 * (-1)^e if e>1, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
%F A136748 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123484.
%F A136748 G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^2 * (1 + x^k + x^(2*k)) * (1 - x^(4*k) + x^(8*k))^2.
%F A136748 Moebius transform is period 24 sequence [ 1, -2, 0, -2, -1, 0, 1, 6, 2, -1, 0, 1, -2, 0, -6, -1, 0, 1, 2, 0, 2, -1, 0, ...].
%F A136748 a(2*n) = A244375(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n).
%F A136748 a(4*n) = -3 * A093829(n). a(4*n + 1) = A112604(n). a(4*n + 2) = -A033762(n). a(4*n + 3) = A112605(n).
%F A136748 a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
%F A136748 Expansion of q * f(-q, -q) * f(q^2, q^10) / f(-q, -q^5)^2 in powers of q where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Oct 12 2015
%F A136748 Sum_{k=1..n} abs(a(k)) ~ (Pi*sqrt(3)/4) * n. - _Amiram Eldar_, Jan 28 2024
%e A136748 G.f. = q - q^2 + q^3 - 3*q^4 - q^6 + 2*q^7 + 3*q^8 + q^9 - 3*q^12 + 2*q^13 + ...
%t A136748 a[ n_] := If[ n < 1, 0, DivisorSum[ n, (Mod[#, 2] - 4 Boole[Mod[#, 8] == 4]) KroneckerSymbol[ -3, n/#] &]]; (* _Michael Somos_, Oct 12 2015 *)
%t A136748 a[ n_] := If[ n < 1, 0, Times @@ (Which[# == 1 || # == 3, 1, # == 2, If[#2 < 2, -1, -3 (-1)^#2], Mod[#, 6] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* _Michael Somos_, Oct 12 2015 *)
%o A136748 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3+A) * eta(x^4 + A)^4 * eta(x^24 + A)^2 / ( eta(x^2 + A) * eta(x^8+A) * eta(x^12+A) )^2, n))};
%o A136748 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, if( e<2, -1, -3 * (-1)^e), p==3, 1, p%6>1, !(e%2), e+1)))};
%o A136748 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, ((d%2) -4 * (d%8==4)) * kronecker(-3, n/d)))};
%Y A136748 Cf. A033762, A093829, A097195, A112604, A112605, A123484.
%Y A136748 Cf. A004016, A005882, A005928.
%K A136748 sign,mult
%O A136748 1,4
%A A136748 _Michael Somos_, Jan 22 2008

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