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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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

R. 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)

M. Somos, Introduction to Ramanujan theta functions

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

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: A135589 A244312 A158122 * A325190 A141416 A176787

Adjacent sequences:  A028638 A028639 A028640 * A028642 A028643 A028644

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 3 20:29 EDT 2020. Contains 333199 sequences. (Running on oeis4.)