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A132001 Expansion of 1 - (1/3) * b(q) * b(q^2) * c(q)^2 / c(q^2) in powers of q where b(), c() are cubic AGM functions.

%I A132001 #24 Feb 16 2025 08:33:06
%S A132001 1,5,1,-11,-24,5,50,53,1,-120,-120,-11,170,250,-24,-203,-288,5,362,
%T A132001 264,50,-600,-528,53,601,850,1,-550,-840,-120,962,821,-120,-1440,
%U A132001 -1200,-11,1370,1810,170,-1272,-1680,250,1850,1320,-24,-2640,-2208,-203,2451,3005
%N A132001 Expansion of 1 - (1/3) * b(q) * b(q^2) * c(q)^2 / c(q^2) in powers of q where b(), c() are cubic AGM functions.
%C A132001 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A132001 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A132001 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.71).
%H A132001 G. C. Greubel, <a href="/A132001/b132001.txt">Table of n, a(n) for n = 1..10000</a>
%H A132001 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132001 Expansion of 1 - phi(-q)^2 * phi(-q^3)^2 * psi(q)^3 / psi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A132001 Expansion of 1 - eta(q) * eta(q^2)^4 * eta(q^3)^5 / eta(q^6)^4 in powers of q.
%F A132001 a(n) is multiplicative with a(2^e) = 2 + ((-4)^(e+1) - 1)/5, a(3^e) = 1, a(p^e) = (q^(e+1) - 1) / (q - 1) where q = p^2 * Kronecker(-3, p) if p > 3.
%F A132001 a(3*n) = a(n).
%F A132001 G.f.: Sum_{k>0} k^2 * Kronecker(-3,k) * x^k / (1 - (-x)^k) = 1 - Product_{k>0} (1 - x^(3k)) * (1 - x^k)^5 / (1 - x^k + x^(2k))^4.
%F A132001 a(n) = - A132000(n) unless n = 0.
%F A132001 Expansion of 1 - (9 * phi(-q) * phi(-q^3)^5 - phi(-q)^5 * phi(-q^3)) / 8 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Nov 02 2015
%F A132001 a(n) = -(-1)^n * A113262(n) unless n = 0. - _Michael Somos_, Nov 02 2015
%e A132001 G.f. = q + 5*q^2 + q^3 - 11*q^4 - 24*q^5 + 5*q^6 + 50*q^7 + 53*q^8 + q^9 - 120*q^10 + ...
%t A132001 a[ n_] := If[ n < 1, 0, -DivisorSum[ n, #^2 (-1)^# KroneckerSymbol[ -3, #] &]]; (* _Michael Somos_, Nov 02 2015 *)
%t A132001 a[ n_] := SeriesCoefficient[ 1 - QPochhammer[ q] QPochhammer[ q^2]^4 QPochhammer[ q^3]^5 / QPochhammer[ q^6]^4, {q, 0, n}]; (* _Michael Somos_, Nov 02 2015 *)
%t A132001 a[ n_] := SeriesCoefficient[ 1 - (1/4) EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^3]^2 EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}]; (* _Michael Somos_, Nov 02 2015 *)
%t A132001 a[ n_] := SeriesCoefficient[ 1 - (9 EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3]^5 - EllipticTheta[ 4, 0, q]^5 EllipticTheta[ 4, 0, q^3]) / 8, {q, 0, n}]; (* _Michael Somos_, Nov 02 2015 *)
%o A132001 (PARI) {a(n) = if(n<1, 0, -sumdiv(n, d, d^2 * (-1)^d * kronecker(-3, d)))};
%o A132001 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 - eta(x + A) * eta(x^2 + A)^4 * eta(x^3 + A)^5 / eta(x^6 + A)^4, n))};
%o A132001 (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==3, 1, p==2, 2 + ((-4)^(e+1) - 1) / 5, p = p^2 * kronecker(-3, p); (p^(e+1) - 1) / (p-1) )))};
%o A132001 (PARI) q='q+O('q^99); Vec(-eta(q)*eta(q^2)^4*eta(q^3)^5/eta(q^6)^4+1) \\ _Altug Alkan_, Sep 07 2018
%Y A132001 Cf. A132000, A113262.
%K A132001 sign,mult
%O A132001 1,2
%A A132001 _Michael Somos_, Aug 06 2007