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A321527 Expansion of x^3 * c(x^2) * c(x^4)^2 / (9 * c(x)) in powers of x where c() is a cubic AGM theta function.

%I A321527 #12 Feb 16 2025 08:33:57
%S A321527 0,0,0,1,-1,0,2,0,-3,4,0,0,1,0,0,6,-7,0,8,0,-6,8,0,0,-1,0,0,13,-8,0,
%T A321527 12,0,-15,12,0,0,7,0,0,14,-18,0,16,0,-12,24,0,0,-5,0,0,18,-14,0,26,0,
%U A321527 -24,20,0,0,6,0,0,32,-31,0,24,0,-18,24,0,0,5,0,0,31
%N A321527 Expansion of x^3 * c(x^2) * c(x^4)^2 / (9 * c(x)) in powers of x where c() is a cubic AGM theta function.
%C A321527 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A321527 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A321527 Number 124 of the 126 eta-quotients listed in Table 1 of Williams 2012.
%H A321527 Antti Karttunen, <a href="/A321527/b321527.txt">Table of n, a(n) for n = 0..16384</a>
%H A321527 Antti Karttunen, <a href="/A321527/a321527.txt">Data supplement: n, a(n) computed for n = 0..100000</a>
%H A321527 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A321527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A321527 K. S. Williams, <a href="http://dx.doi.org/10.1142/S1793042112500595">Fourier series of a class of eta quotients</a>, Int. J. Number Theory 8 (2012), no. 4, 993-1004.
%F A321527 Expansion of x^3 * (psi(x^3) * psi(x^6))^3 / (psi(x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function.
%F A321527 Expansion of x^3 * chi(-x) * f(-x^12)^6 / (chi(-x^3)^3 * f(-x^4)^2) in powers of x where chi(), f() are Ramanujan theta functions.
%F A321527 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321528.
%F A321527 a(n) = s(n/3) - s(n/4) - s(n/6) + s(n/12) where s(x) = sum of divisors of x for integer x else 0.
%F A321527 a(6*n + 1) = a(6*n + 5) = a(12*n + 2) = a(12*n + 10) = 0.
%F A321527 a(n) = A224226(n) if n>0. a(2*n) = -A229615(n). a(6*n + 3) = A008438(n). a(12*n + 6) = 2*A008438(n).
%F A321527 a(12*n + 3) = A112610(n). a(12*n + 4) = -A144614(n). a(12*n + 8) = -3*A033686(n). a(12*n + 9) = 4*A097723(n).
%e A321527 G.f. = x^3 - x^4 + 2*x^6 - 3*x^8 + 4*x^9 + x^12 + 6*x^15 - 7*x^16 + ...
%t A321527 a[ n_] := SeriesCoefficient[ x^3 QPochhammer[ x, x^2] QPochhammer[ x^12]^6 / (QPochhammer[ x^3, x^6]^3 QPochhammer[ x^4]^2), {x, 0, n}];
%t A321527 a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, 0, s[n/3] - s[n/4] - s[n/6] + s[n/12]]];
%t A321527 a[ n_] := If[ n < 1, 0, Sum[ d {0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0, 0}[[Mod[d, 12, 1]]] / 12, {d, Divisors[n]}]];
%o A321527 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0][d%12 + 1] / 12))};
%o A321527 (PARI) {a(n) = my(s = x -> if( frac(x), 0, sigma(x))); if( n<1, 0, s(n/3) - s(n/4) - s(n/6) + s(n/12))};
%o A321527 (PARI) {a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 * eta(x^12 + A)^6 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^4 + A)^2), n))};
%o A321527 (Magma) A := Basis( ModularForms( Gamma0(12), 2), 76); A[4] - A[5];
%Y A321527 Cf. A008438, A033686, A097723, A112610, A144614, A224226, A229615, A321528.
%K A321527 sign
%O A321527 0,7
%A A321527 _Michael Somos_, Nov 12 2018