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%I #22 Feb 02 2024 11:02:01
%S 1,2,-2,-4,1,-4,-6,8,7,2,12,8,-12,-12,-2,-16,-16,14,20,-4,12,24,-22,
%T -16,1,-24,-20,24,30,-4,32,32,-24,-32,-6,-28,-36,40,24,8,42,24,-42,
%U -48,7,-44,-46,32,43,2,32,48,-52,-40,12,-48,-40,60,60,8,62,64,-42
%N Expansion of eta(q^2)^7 * eta(q^5)^2 * eta(q^20)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^10)^3) in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A138558/b138558.txt">Table of n, a(n) for n = 1..1000</a>
%H L.-C. Shen, <a href="https://doi.org/10.1090/S0002-9947-1994-1250827-3">On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5</a>, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345. See p. 343, Eq. (3.44).
%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 q * (f(q)^3 / chi(q)) * (f(q^5) / chi(q^5)^3) in powers of q where chi(), f() are Ramanujan theta functions.
%F Expansion of ( f(q)^5 / f(q^5) - f(-q^2)^5 / f(-q^10) ) / 5 in powers of q where f() is a Ramanujan theta function.
%F Euler transform of period 20 sequence [ 2, -5, 2, -3, 0, -5, 2, -3, 2, -4, 2, -3, 2, -5, 0, -3, 2, -5, 2, -4, ...].
%F a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, a(5^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 2, 3 (mod 5).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138557.
%F G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k))^3 * (1 - x^(5*k)) * (1 - x^(10*k-5)) * (1 + x^(10*k))^2.
%F G.f.: Sum_{k>0} (-1)^k * ( f(5*k-1) + f(5*k-2) - f(5*k-3) - f(5*k-4) ) where f(k) := k * x^k / (1 - x^(2*k)).
%F a(n) = -(-1)^n * A111580(n). a(2*n) = 2 * A111580(n). - _Michael Somos_, Sep 08 2015
%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . - _Amiram Eldar_, Feb 01 2024
%e G.f. = q + 2*q^2 - 2*q^3 - 4*q^4 + q^5 - 4*q^6 - 6*q^7 + 8*q^8 + 7*q^9 + 2*q^10 + ...
%t a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, # Mod[n/#, 2] KroneckerSymbol[ 5, #] &]]; (* _Michael Somos_, Sep 08 2015 *)
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -q]^5 / QPochhammer[ -q^5] - QPochhammer[ q^2]^5 / QPochhammer[ q^10]) / 5, {q, 0, n}]; (* _Michael Somos_, Sep 08 2015 *)
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ -q]^3 Pochhammer[ q, -q] QPochhammer[ -q^5] QPochhammer[ q^5, -q^5]^3, {q, 0, n}]; (* _Michael Somos_, Sep 08 2015 *)
%o (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, (n/d%2) * d * kronecker(5, d)))};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^5 + A)^2 * eta(x^20 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^3), n))};
%Y Cf. A111580, A138557.
%Y Cf. A000122, A000700, A010054, A121373.
%K sign,mult
%O 1,2
%A _Michael Somos_, Mar 24 2008, Mar 25 2008
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