A028641 Expansion of theta_3(q) * theta_3(q^19) + theta_2(q) * theta_2(q^19) in powers of q.

1, 2, 0, 0, 2, 4, 0, 4, 0, 2, 0, 4, 0, 0, 0, 0, 2, 4, 0, 2, 4, 0, 0, 4, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 0, 4, 4, 4, 0, 4, 0, 6, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 4, 0, 4, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 2, 8, 0, 0, 4, 2, 0, 4, 0, 8, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 6, 4, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

The number of integer solutions (x, y) to x^2 + x*y + 5*y^2 = n, discriminant -19. - Ray Chandler, Jul 12 2014

Examples

			G.f. = 1 + 2*x + 2*x^4 + 4*x^5 + 4*x^7 + 2*x^9 + 4*x^11 + 2*x^16 + 4*x^17 + 2*x^19 + ...
		

References

• Robert Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 409, Eq. (19).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A035171.

Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), this sequence (d=-19), A138811 (d=-43).

Programs

Mathematica

a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -19, #] &] 2]; (* Michael Somos, Jun 14 2012 *)

PARI

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-19, d)) * 2)}; /* Michael Somos, Feb 27 2007 */

PARI

{a(n) = if( n<1, n==0, qfrep([2, 1;1, 10], n, 1)[n] * 2)}; /* Michael Somos, Feb 27 2007 */

Formula

Theta series of quadratic form with Gram matrix [ 2, 1; 1, 10 ].

Expansion of phi(q) * phi(q^19) + 4 * q^5 * psi(q^2)* psi(q^38) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 27 2007

Moebius transform is period 19 sequence [2, -2, -2, 2, 2, 2, 2, -2, 2, -2, 2, -2, -2, -2, -2, 2, 2, -2, 0, ...]. - Michael Somos, Feb 27 2007

a(n) = 2*b(n) where b(n) is multiplicative with a(0) = 1, b(19^e) = 1, b(p^e) = e + 1 if Kronecker(-19, p) = 1, b(p^e) = (1 + (-1)^e)/2 if Kronecker(-19, p) = -1. - Michael Somos, Feb 27 2007

a(n) = 2 * A035171(n) unless n = 0. - Jianing Song, Sep 06 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(19) = 1.441461... . - Amiram Eldar, Dec 16 2023