A138805 Theta series of quadratic form x^2 + x*y + 7*y^2.

1, 2, 0, 0, 2, 0, 0, 4, 0, 6, 0, 0, 0, 4, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 6, 4, 0, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 12, 2, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 6, 0, 0, 0, 0, 0

Offset

0,2

Comments

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

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 theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) in powers of q.

Expansion of phi(q) * phi(q^27) + 4 * q^7 * psi(q^2) * psi(q^54) in powers of q where phi(), psi() are Ramanujan theta functions.

Moebius transform is period 27 sequence [ 2, -2, -2, 2, -2, 2, 2, -2, 6, 2, -2, -2, 2, -2, 2, 2, -2, -6, 2, -2, -2, 2, -2, 2, 2, -2, 0, ...].

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 27^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

G.f.: Sum_{i, j in Z} x^(i*i + i*j + 7*j*j).

a(3*n + 2) = a(4*n + 2) = 0.

a(n) = 2 * A138806(n) unless n=0. a(9*n) = A004016(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Dec 29 2023

Example

G.f. = 1 + 2*q + 2*q^4 + 4*q^7 + 6*q^9 + 4*q^13 + 2*q^16 + 4*q^19 + 2*q^25 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -3, n/#] {1, 1, 0, 1, 1, 0, 1, 1, 3}[[Mod[#, 9, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^27] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^27], {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 14], n, 1)), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^54 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^27 + A) * eta(x^108 + A))^2 + 4 * x^7 * (eta(x^4 + A) * eta(x^108 + A))^2 / (eta(x^2 + A) * eta(x^54 + A)), n))};
(PARI) {a(n) = if( n<1, n==0, 2 * sumdiv(n, d, kronecker(-3, n/d) * [ 3, 1, 1, 0, 1, 1, 0, 1, 1][n%9 + 1]))}; /* Michael Somos, Sep 08 2015 */
(Magma) A := Basis( ModularForms( Gamma1(27), 1), 87); A[1] + 2*A[2] + 2*A[5] + 4*A[8] + 6*A[10] + 4*A[14] + 2*A[15]; /* Michael Somos, Sep 08 2015 */

Crossrefs

Cf. A004016, A138806, A248897.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A171608 A307985 A024164 * A316400 A061897 A372568

Adjacent sequences: A138802 A138803 A138804 * A138806 A138807 A138808

Keyword

nonn,easy

Author

Michael Somos, Mar 30 2008

Status

approved