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A282544 Expansion of (phi(x)^4 + 3*phi(x^3)^4) / 4 in powers of x where phi() is a Ramanujan theta function.

%I A282544 #34 Mar 08 2025 17:15:42
%S A282544 1,2,6,14,6,12,42,16,6,50,36,24,42,28,48,84,6,36,150,40,36,112,72,48,
%T A282544 42,62,84,158,48,60,252,64,6,168,108,96,150,76,120,196,36,84,336,88,
%U A282544 72,300,144,96,42,114,186,252,84,108,474,144,48,280,180,120,252
%N A282544 Expansion of (phi(x)^4 + 3*phi(x^3)^4) / 4 in powers of x where phi() is a Ramanujan theta function.
%C A282544 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A282544 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A282544 a(n) is the number of solutions in integers to n = x^2 + y^2 + z^2 + w^2 where x + y + z = 3m is a multiple of 3. - _Michael Somos_, Jun 23 2018
%H A282544 G. C. Greubel, <a href="/A282544/b282544.txt">Table of n, a(n) for n = 0..10000</a>
%H A282544 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A282544 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A282544 Expansion of a(x^2) * phi(x) * phi(x^3) in powers of x where a() is a cubic AGM theta function and phi() is a Ramanujan theta function.
%F A282544 Expansion of (chi(x) * chi(x^3))^3 * (psi(x)^4 + 3*x*psi(x^3)^4) in powers of x where psi(), chi() are Ramanujan theta functions.
%F A282544 a(n) = 2*b(n) where b() is multiplicative with a(0) = 1, b(2^e) = 3 if e>0, b(3^e) = 3^(e+1) - 2, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
%F A282544 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i)^2 f(t) where q = exp(2 Pi i t).
%F A282544 G.f.: ((Sum_{k in Z} x^k^2)^4 + 3 * (Sum_{k in Z} x^(3*k^2))^4) / 4.
%F A282544 G.f.: 1 + 2 * Sum_{k>0} F(k, x) + 6 * Sum_{k>0} F(3*k, x) where F(k, x) = x^k / (1 + (-x)^k)^2.
%F A282544 G.f.: 1 + 2 * Sum_{k>0} F(k, x) + 2 * Sum_{k>0} F(3*k, x) where F(k, x) = k * x^k / (1 + (-x)^k).
%F A282544 a(2*n) = A125510(n). a(n) = A033712(2*n).
%F A282544 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/6 = 1.644934... (A013661). - _Amiram Eldar_, Dec 29 2023
%e A282544 G.f. = 1 + 2*x + 6*x^2 + 14*x^3 + 6*x^4 + 12*x^5 + 42*x^6 + 16*x^7 + 6*x^8 + ...
%e A282544 a(4) = 6 with solutions (x, y, z, w) = {(1, 1, 1, 1), (1, 1, 1, -1), (0, 0, 0, 2)} and their negatives. - _Michael Somos_, Jun 23 2018
%t A282544 a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[n, # {1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 0}[[Mod[#, 12, 1]]] &]];
%t A282544 a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ (Which[# < 3, 2 + (-1)^#, # == 3, 3^(#2 + 1) - 2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger@n)];
%t A282544 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^4 + 3 * EllipticTheta[ 3, 0, x^3]^4) / 4, {x, 0, n}];
%o A282544 (PARI) {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, d * [0, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1][d%12+1]))};
%o A282544 (PARI) {a(n) = if( n<1, n==0, my(A = factor(n), p, e); 2 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3, p==3, 3^(e+1) - 2, (p^(e+1) - 1) / (p - 1))))};
%o A282544 (PARI) {a(n) = if( n<0, 0, my(A); A = x * O(x^n); polcoeff( (sum(k=1, sqrtint(n), 2 * x^k^2, 1 + A)^4 + 3 * sum(k=1, sqrtint(n\3), 2 * x^(3*k^2), 1 + A)^4) / 4, n))};
%o A282544 (Magma) A := Basis( ModularForms( Gamma0(12), 2), 60); A[1] + 2*A[2] + 6*A[3] + 14*A[4] + 6*A[5];
%Y A282544 Cf. A013661, A033712, A125510.
%Y A282544 Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.
%K A282544 nonn,easy
%O A282544 0,2
%A A282544 _Michael Somos_, Feb 17 2017