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A028641
Expansion of theta_3(q) * theta_3(q^19) + theta_2(q) * theta_2(q^19) in powers of q.
10
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
0,2
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
REFERENCES
Robert Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 409, Eq. (19).
LINKS
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
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
EXAMPLE
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 + ...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -19, #] &] 2]; (* Michael Somos, Jun 14 2012 *)
PROG
(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 */
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).
Sequence in context: A158122 A348165 A366531 * A325190 A141416 A176787
KEYWORD
nonn,easy
STATUS
approved