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A214304
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Expansion of phi(q) + phi(q^2) - phi(q^4) in powers of q where phi() is a Ramanujan theta function.
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0
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1, 2, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0
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OFFSET
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0,2
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COMMENTS
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Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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LINKS
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Table of n, a(n) for n=0..86.
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
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Expansion of psi(-x^2) * phi(x^4)^2 / f(-x, x^3)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 16 sequence [ 2, -1, -2, 2, -2, 1, 2, -3, 2, 1, -2, 2, -2, -1, 2, -1, ...].
a(n) = 2 * b(n) where b(n) is multiplicative with b(2^e) = (1 - (-1)^e)/2, b(p^e) = (1 + (-1)^e)/2 if p>2.
G.f.: Sum_{k} x^k^2 + x^(2*k^2) - x^(4*k^2).
a(4*n + 3) = a(6*n + 4) = a(6*n + 5) = a(8*n + 4) = a(8*n + 5) = a(8*n + 6) = 0. a(2*n) = A000122(n). a(6*n + 2) = 2 * A089801(n). a(8*n + 1) = 2 * A010054(n).
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EXAMPLE
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1 + 2*q + 2*q^2 + 2*q^8 + 2*q^9 + 2*q^18 + 2*q^25 + 2*q^32 + 2*q^49 + ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2] - EllipticTheta[ 3, 0, q^4], {q, 0, n}]
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, I q] / (2 (I q)^(1/4)) EllipticTheta[ 3, 0, q^4]^2 / (QPochhammer[ -q^4] QPochhammer[ q, -q^4] QPochhammer[ -q^3, -q^4])^2, {q, 0, n}]
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PROG
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(PARI) {a(n) = if( n<1, n==0, 2 * issquare( n * 2^(n%2==0)))}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, (1 - (-1)^e)/2, (1 + (-1)^e)/2))))}
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CROSSREFS
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Cf. A000122, A010054, A089801.
Sequence in context: A191928 A033148 A186230 * A027359 A035447 A037863
Adjacent sequences: A214301 A214302 A214303 * A214305 A214306 A214307
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Jul 12 2012
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STATUS
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approved
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