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

0,2

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

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.

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

Example

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

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}];

Prog

(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, p , e); A = factor(n); 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))};
(Magma) A := Basis( ModularForms( Gamma0(12), 2), 60); A[1] + 2*A[2] + 6*A[3] + 14*A[4] + 6*A[5];

Crossrefs

Cf. A013661, A033712, A125510.

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

Sequence in context: A266960 A056592 A349681 * A091719 A058054 A054588

Adjacent sequences: A282541 A282542 A282543 * A282545 A282546 A282547

Keyword

nonn,easy

Author

Michael Somos, Feb 17 2017

Status

approved