A138805 - OEIS

%I #18 Dec 29 2023 03:49:21

%S 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,

%T 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,

%U 0,0,0,0,0,4,0,0,4,0,0,4,0,6,0,0,0,0,0

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

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

%H G. C. Greubel, <a href="/A138805/b138805.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

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

%F 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.

%F 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, ...].

%F 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).

%F 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).

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

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

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

%F 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

%e 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 + ...

%t 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 *)

%t 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 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 14], n, 1)), n))};

%o (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))};

%o (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 */

%o (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 */

%Y Cf. A004016, A138806, A248897.

%Y Cf. A000122, A000700, A010054, A121373.

%K nonn,easy

%O 0,2

%A _Michael Somos_, Mar 30 2008