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%I #22 Nov 24 2023 08:13:21
%S 1,2,0,0,2,0,0,0,1,4,0,0,2,0,0,0,2,2,0,0,0,0,0,0,3,4,0,0,2,0,0,0,0,4,
%T 0,0,2,0,0,0,2,0,0,0,2,0,0,0,1,6,0,0,2,0,0,0,0,4,0,0,2,0,0,0,4,0,0,0,
%U 0,0,0,0,2,4,0,0,0,0,0,0,1,4,0,0,4,0,0
%N Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A134013/b134013.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of eta(q^2)^5 * eta(q^16)^2 / ( eta(q)^2 * eta(q^4)^2 * eta(q^8) ) in powers of q.
%F Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, ...].
%F Moebius transform is period 16 sequence [ 1, 1, -1, -2, 1, -1, -1, 0, 1, 1, -1, 2, 1, -1, -1, 0, ...].
%F a(n) is multiplicative with a(2) = 2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134014.
%F a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = 2 * a(4*n + 1).
%F G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k)^2 / (1 - x^(4*k)).
%F a(n) = -(-1)^n * A112301(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n = 5) = 2 * A053692(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 (A003881). - _Amiram Eldar_, Nov 24 2023
%e G.f. = q + 2*q^2 + 2*q^5 + q^9 + 4*q^10 + 2*q^13 + 2*q^17 + 2*q^18 + 3*q^25 + ...
%t a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^4], {q, 0, n}]; (* _Michael Somos_, Oct 30 2015 *)
%o (PARI) {a(n) = if( n>0 && (n+1)%4\2, (n%4) * sumdiv( n/gcd(n,2), d, (-1)^(d\2)))};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^8 + A)), n))};
%Y Cf. A003881, A008441, A053692, A112301, A113407, A134014.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy,mult
%O 1,2
%A _Michael Somos_, Oct 02 2007
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