A214304 Expansion of phi(q) + phi(q^2) - phi(q^4) in powers of q where phi() is a Ramanujan theta function.

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

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

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).

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = sqrt(2) + 1 = 2.414213... (A014176). - Amiram Eldar, Dec 30 2023

Example

1 + 2*q + 2*q^2 + 2*q^8 + 2*q^9 + 2*q^18 + 2*q^25 + 2*q^32 + 2*q^49 + ...

Mathematica

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}]

Prog

(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))))}

Crossrefs

Cf. A000122, A000700, A010054, A014176, A089801, A121373.

Sequence in context: A033148 A281084 A186230 * A248640 A376562 A281083

Adjacent sequences: A214301 A214302 A214303 * A214305 A214306 A214307

Keyword

nonn,easy

Author

Michael Somos, Jul 12 2012

Status

approved