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A132000 Expansion of (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 A132000 #17 Feb 16 2025 08:33:06
%S A132000 1,-1,-5,-1,11,24,-5,-50,-53,-1,120,120,11,-170,-250,24,203,288,-5,
%T A132000 -362,-264,-50,600,528,-53,-601,-850,-1,550,840,120,-962,-821,120,
%U A132000 1440,1200,11,-1370,-1810,-170,1272,1680,-250,-1850,-1320,24,2640,2208,203,-2451
%N A132000 Expansion of (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 A132000 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A132000 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A132000 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.71).
%H A132000 G. C. Greubel, <a href="/A132000/b132000.txt">Table of n, a(n) for n = 0..10000</a>
%H A132000 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132000 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132000 Expansion of 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 A132000 Expansion of eta(q) * eta(q^2)^4 * eta(q^3)^5 / eta(q^6)^4 in powers of q.
%F A132000 Euler transform of period 6 sequence [-1, -5, -6, -5, -1, -6, ...].
%F A132000 a(n) = -b(n) where b() is multiplicative with b(2^e) = 2+((-4)^(e+1)-1)/5, b(3^e) = 1, b(p^e) = (q^(e+1) - 1) / (q-1) where q = p^2*Kronecker(-3, p) if p > 3.
%F A132000 a(3*n) = a(n).
%F A132000 G.f.: 1 - Sum_{k>0} k^2 * Kronecker(-3, k) * x^k / (1 - (-x)^k) = Product_{k>0} (1  - x^(3*k)) * (1 - x^k)^5 / (1 - x^k + x^(2*k))^4.
%F A132000 a(n) = (-1)^n * A113261(n). Convolution of A123330 and A131943.
%F A132000 a(n) = -A132000(n) unless n=0.
%F A132000 Expansion of (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 03 2015
%F A132000 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 15552^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122373. - _Michael Somos_, Nov 03 2015
%e A132000 G.f. = 1 - x - 5*x^2 - x^3 + 11*x^4 + 24*x^5 - 5*x^6 - 50*x^7 - 53*x^8 - x^9 + ...
%t A132000 a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, #^2 (-1)^# KroneckerSymbol[ -3, #] &]]; (* _Michael Somos_, Nov 03 2015 *)
%t A132000 a[ n_] := SeriesCoefficient[ (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 03 2015 *)
%t A132000 a[ n_] := SeriesCoefficient[(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 03 2015 *)
%t A132000 a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^2]^4 QPochhammer[ q^3]^5 / QPochhammer[ q^6]^4, {q, 0, n}]; (* _Michael Somos_, Nov 03 2015 *)
%o A132000 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, d^2 * (-1)^d * kronecker(-3, d)))};
%o A132000 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^4 * eta(x^3 + A)^5 / eta(x^6 + A)^4, n))};
%o A132000 (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==3, 1, p==2, 2 + ((-4)^(e+1) - 1) / 5, p = p^2 * kronecker(-3, p); (p^(e+1) - 1) / (p-1) )))};
%o A132000 (Magma) A := Basis( ModularForms( Gamma1(6), 3), 50); A[1] - A[2] - 5*A[3] - A[4]; /* _Michael Somos_, Nov 03 2015 */
%Y A132000 Cf. A113261, A122373, A123330, A131943.
%K A132000 sign
%O A132000 0,3
%A A132000 _Michael Somos_, Aug 06 2007