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A260671 Expansion of theta_3(q) * theta_3(q^15) in powers of q. 5
1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 10, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is the number of solutions in integers (x, y) of x^2 + 15*y^2 = n. - Michael Somos, Jul 17 2018

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

FORMULA

Expansion of (eta(q^2) * eta(q^30))^5 / (eta(q) * eta(q^4) * eta(q^15) * eta(q^60))^2 in powers of q.

Euler transform of a period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = 60^(1/2) (t/i) f(t) where q = exp(2 Pi i t).

G.f.: (Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(15*k^2)).

a(3*n + 2) = a(4*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.

a(4*n) = A028625(n). a(4*n + 1) = 2 * A260675(n). a(4*n + 3) = 2 * A260676(n).

a(5*n) = A192323(n).

a(n) = A122855(n) + A140727(n).

EXAMPLE

G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^15 + 6*x^16 + 4*x^19 + 4*x^24 + 2*x^25 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^15], {q, 0, n}];

PROG

(PARI) {a(n) = if( n<1, n==0, qfrep([1, 0; 0, 15], n)[n]*2)};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^15 + A) * eta(x^60 + A))^2, n))};

(PARI) q='q+O('q^99); Vec((eta(q^2)*eta(q^30))^5/(eta(q)*eta(q^4)*eta(q^15)*eta(q^60))^2) \\ Altug Alkan, Jul 18 2018

CROSSREFS

Cf. A028625, A122855, A140727, A192323, A260675, A260676.

Sequence in context: A227395 A255258 A329266 * A033725 A204010 A033723

Adjacent sequences:  A260668 A260669 A260670 * A260672 A260673 A260674

KEYWORD

nonn

AUTHOR

Michael Somos, Nov 14 2015

STATUS

approved

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Last modified April 17 08:34 EDT 2021. Contains 343064 sequences. (Running on oeis4.)