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A033719 Coefficients in expansion of theta_3(q) * theta_3(q^7) in powers of q. 7
1, 2, 0, 0, 2, 0, 0, 2, 4, 2, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 4, 0, 0, 8, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 4, 0, 0, 0, 4, 4, 0, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of integer solutions to the equation x^2 + 7*y^2 = n.

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.

FORMULA

Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2 + 7*j^2).

Euler transform of period 28 sequence [ 2, -3, 2, -1, 2, -3, 4, -1, 2, -3, 2, -1, 2, -6, 2, -1, 2, -3, 2, -1, 4, -3, 2, -1, 2, -3, 2, -2, ...].

Expansion of (eta(q^2) * eta(q^14))^5 / (eta(q) * eta(q^4) * eta(q^7) * eta(q^28))^2 in powers of q.

G.f.: Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^2 * (1 - x^(14*k)) * (1 + x^(14*k-7))^2.

EXAMPLE

G.f. = 1 + 2*x + 2*x^4 + 2*x67 + 4*x^8 + 2*x^9 + 4*x^11 + 6*x^16 + 4*x^23 + ...

MAPLE

seq(coeff(series(mul((1-x^(2*k))*(1+x^(2*k-1))^2*(1-x^(14*k))*(1+x^(14*k-7))^2, k=1..n), x, n+1), x, n), n = 0 .. 110); # Muniru A Asiru, Feb 02 2019

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^7], {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-2 * eta(x^7 + A)^-2 * eta(x^14 + A)^5 * eta(x^28 + A)^-2, n))};

CROSSREFS

Sequence in context: A216921 A344982 A280285 * A171608 A307985 A024164

Adjacent sequences:  A033716 A033717 A033718 * A033720 A033721 A033722

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 29 10:40 EDT 2021. Contains 346344 sequences. (Running on oeis4.)