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

1, 2, 6, 14, 6, 12, 42, 16, 6, 50, 36, 24, 42, 28, 48, 84, 6, 36, 150, 40, 36, 112, 72, 48, 42, 62, 84, 158, 48, 60, 252, 64, 6, 168, 108, 96, 150, 76, 120, 196, 36, 84, 336, 88, 72, 300, 144, 96, 42, 114, 186, 252, 84, 108, 474, 144, 48, 280, 180, 120, 252

Offset: 0

Author

Michael Somos, Feb 17 2017

Keywords

• nonn
• easy

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

Examples

			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 + ...
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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A013661, A033712, A125510.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Programs

Magma

A := Basis( ModularForms( Gamma0(12), 2), 60); A[1] + 2*A[2] + 6*A[3] + 14*A[4] + 6*A[5];

Mathematica

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]]] &]];
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)];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^4 + 3 * EllipticTheta[ 3, 0, x^3]^4) / 4, {x, 0, n}];

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]))};

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))))};

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))};

Formula

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.

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.

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.

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).

G.f.: ((Sum_{k in Z} x^k^2)^4 + 3 * (Sum_{k in Z} x^(3*k^2))^4) / 4.

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.

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).

a(2*n) = A125510(n). a(n) = A033712(2*n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/6 = 1.644934... (A013661). - Amiram Eldar, Dec 29 2023