A033762 Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).

1, 1, 0, 2, 1, 0, 2, 0, 0, 2, 2, 0, 1, 1, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 4, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 2, 0

Offset

0,4

Comments

Number of solutions of 8*n + 4 = x^2 + 3*y^2 in positive odd integers. - Michael Somos, Sep 18 2004

Half the number of integer solutions of 4*n + 2 = x^2 + y^2 + z^2 where 0 = x + y + z and x and y are odd. - Michael Somos, Jul 03 2011

Given g.f. A(x), then q^(1/2) * 2 * A(q) is denoted phi_1(z) where q = exp(Pi i z) in Conway and Sloane.

Half of theta series of planar hexagonal lattice (A2) with respect to an edge.

Bisection of A002324. Number of ways of writing n as a sum of a triangular plus three times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011

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).

References

Burce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 223 Entry 3(i).

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 103. See Eq. (13).

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.27).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Michael D. Hirschhorn, Three classical results on representations of a number, Sem. Lotharingien de Combinat. S42 (1999), B42f.

Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

Michael Somos, Introduction to Ramanujan theta functions, 2010.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

Expansion of q^(-1/2) * (eta(q^2) * eta(q^6))^2 / (eta(q) * eta(q^3)) in powers of q. - Michael Somos, Apr 18 2004

Expansion of q^(-1) * (a(q) - a(q^4)) / 6 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Oct 24 2006

Expansion of psi(x) * psi(x^3) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 03 2011

Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, -2, ...]. - Michael Somos, Apr 18 2004

From Michael Somos, Sep 18 2004: (Start)

Given g.f. A(x), then B(x) = (x * A(x^2))^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 4*u*v*w + 16*v*w^2 - 8*w*v^2 - w*u^2.

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p==5 (mod 6) otherwise b(p^e) = e+1. (Clarification: the g.f. A(x) is not the primary function of interest, but rather B(x) = x * A(x^2), which is an eta-quotient and is the generating function of a multiplicative sequence.)

G.f.: (Sum_{j>0} x^((j^2 - j) / 2)) * (Sum_{k>0} x^(3(k^2 - k) / 2)) = Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)).

G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k * (1 - x^k) * (1 - x^(4*k)) * (1 - x^(5*k)) / (1 - x^(12*k)). (End)

G.f.: s(4)^2*s(12)^2/(s(2)*s(6)), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]

G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - x^(4*k) / (1 + x^(4*k) + x^(8*k)). - Michael Somos, Nov 04 2005

a(n) = A002324(2*n + 1) = A035178(2*n + 1) = A091393(2*n + 1) = A093829(2*n + 1) = A096936(2*n + 1) = A112298(2*n + 1) = A113447(2*n + 1) = A113661(2*n + 1) = A113974(2*n + 1) = A115979(2*n + 1) = A122860(2*n + 1) = A123331(2*n + 1) = A123484(2*n + 1) = A136748(2*n + 1) = A137608(2*n + 1). A005881(n) = 2*a(n).

6 * a(n) = A004016(6*n + 3). - Michael Somos, Mar 06 2016

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 23 2023

Example

G.f. = 1 + x + 2*x^3 + x^4 + 2*x^6 + 2*x^9 + 2*x^10 + x^12 + x^13 + 2*x^15 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 2*q^13 + 2*q^19 + 2*q^21 + q^25 + q^27 + 2*q^31 + ...
a(6) = 2 since 8*6 + 4 = 52 = 5^2 + 3*3^2 = 7^2 + 3*1^2.

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Jul 03 2011 *)
QP = QPochhammer; s = (QP[q^2]*QP[q^6])^2/(QP[q]*QP[q^3]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(2 n + 1))]; (* Michael Somos, Mar 06 2016 *)
%t A033762 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}]; (* Michael Somos, Mar 06 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 18 2004 */
(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -12, d) * (n / d % 2)))}; /* Michael Somos, Nov 04 2005 */
(PARI) {a(n) = if( n<0, 0, n = 8*n + 4; sum( j=1, sqrtint( n\3), (j%2) * issquare(n - 3*j^2)))} /* Michael Somos, Nov 04 2005 */
(PARI) {a(n) = if( n<0, 0, sumdiv(2*n + 1, d, kronecker(-3, d)))}; /* Michael Somos, Mar 06 2016 */
(Magma) A := Basis( ModularForms( Gamma1(12), 1), 202); A[2] + A[4]; /* Michael Somos, Jul 25 2014 */

Crossrefs

Cf. A002324, A004016, A005881, A035178, A091393, A093766, A093829, A096936, A112298, A113447, A113661, A113974, A115979, A122860, A123331, A123484, A136748, A137608.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A112608 A058677 A262780 * A129449 A033798 A033792

Adjacent sequences: A033759 A033760 A033761 * A033763 A033764 A033765

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Corrected by Charles R Greathouse IV, Sep 02 2009

Status

approved