A227354 Expansion of 2 * a(q) - a(q^2) in powers of q where a() is a cubic AGM theta function.

1, 12, -6, 12, 12, 0, -6, 24, -6, 12, 0, 0, 12, 24, -12, 0, 12, 0, -6, 24, 0, 24, 0, 0, -6, 12, -12, 12, 24, 0, 0, 24, -6, 0, 0, 0, 12, 24, -12, 24, 0, 0, -12, 24, 0, 0, 0, 0, 12, 36, -6, 0, 24, 0, -6, 0, -12, 24, 0, 0, 0, 24, -12, 24, 12, 0, 0, 24, 0, 0, 0

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).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Expansion of (4 * b(q^4)^2 - 2 * b(q) * b(q^4) - b(q)^2) / b(q^2) in powers of q where b() is a cubic AGM theta function.

Expansion of phi(-q^2)^3 / phi(-q^6) + 12 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jan 09 2015

Expansion of theta_4(q^2)^3 / theta_4(q^6) + 3 * theta_2(q) * theta_2(q^3) in powers of q.

Moebius transform is period 6 sequence [ 12, -18, 0, 18, -12, 0, ...].

a(n) = 12 * b(n) where b(n) is multiplicative with b(2^e) = (1 + 3*(-1)^e) / 4, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).

a(n) = A122859(8*n). a(2*n) = A122859(n). a(2*n + 1) = 12 * A033762(n). a(4*n) = a(n). a(4*n + 1) = 12 * A112604(n). a(4*n + 2) = -6 * A033762(n). a(4*n + 3) = 12 * A112605(n).

G.f.: 1 + 6 * Sum_{k>0} (mod(k, 2) + 1) * x^k / (1 + x^k + x^(2*k)). - Michael Somos, Jan 09 2015

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3) = 5.441398... (A304656). - Amiram Eldar, Nov 23 2023

Example

G.f. = 1 + 12*q - 6*q^2 + 12*q^3 + 12*q^4 - 6*q^6 + 24*q^7 - 6*q^8 + 12*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2]^3 / EllipticTheta[ 4, 0, q^6] + 3 EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 6 Sum[ JacobiSymbol[ d, 3] (Mod[ n/d, 2] + 1), {d, Divisors@n}]]; (* Michael Somos, Jan 09 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 12 * sumdiv( n, d, kronecker( d, 3)) - 6 * sumdiv( 2*n, d, kronecker( d, 3)))};
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 12 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, (1 + 3*(-1)^e) / 4, if( p == 3, 1, if( p%6 == 1, e+1, (1 + (-1)^e) / 2 ))))))};

Crossrefs

Cf. A033762, A112604, A112605, A122859, A304656.

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

Sequence in context: A283880 A084067 A240537 * A328043 A075247 A257841

Adjacent sequences: A227351 A227352 A227353 * A227355 A227356 A227357

Keyword

sign,easy

Author

Michael Somos, Jul 08 2013

Status

approved