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A329290
Number of ordered triples (i, j, k) of integers such that n = i^2 + 4*j^2 + 4*k^2.
0
1, 2, 0, 0, 6, 8, 0, 0, 12, 10, 0, 0, 8, 8, 0, 0, 6, 16, 0, 0, 24, 16, 0, 0, 24, 10, 0, 0, 0, 24, 0, 0, 12, 16, 0, 0, 30, 8, 0, 0, 24, 32, 0, 0, 24, 24, 0, 0, 8, 18, 0, 0, 24, 24, 0, 0, 48, 16, 0, 0, 0, 24, 0, 0, 6, 32, 0, 0, 48, 32, 0, 0, 36, 16, 0, 0, 24, 32
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(5/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A169783.
LINKS
FORMULA
Euler transform of period 16 sequence [2, -3, 2, 3, 2, -3, 2, -7, 2, -3, 2, 3, 2, -3, 2, -3, ...].
Expansion of phi(x) * phi(x^4)^2 = phi(x^4)^3 + 2*x*phi(x^4)*psi(x^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
G.f.: theta_3(q) * theta_3(q^4)^2, where theta_3() is the Jacobi theta function.
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = a(4*n + 3) = 0.
EXAMPLE
G.f. = 1 + 2*x + 6*x^4 + 8*x^5 + 12*x^8 + 10*x^9 + 8*x^12 + 8*x^13 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^4]^2, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^10 / (eta(x + A)^2 * eta(x^4 + A)^6 * eta(x^16 + A)^4), n))};
(Magma) A := Basis( ModularForms( Gamma0(16), 3/2), 77); A[1] + 2*A[2];
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Nov 17 2019
STATUS
approved