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A138557 Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^10)^7 / (eta(q^2)^3 * eta(q^5)^2 * eta(q^20)^2) in powers of q.

%I A138557 #16 Feb 16 2025 08:33:08
%S A138557 1,-2,2,-4,5,-4,6,-8,7,-10,12,-8,12,-12,10,-16,16,-14,20,-20,12,-24,
%T A138557 22,-16,25,-24,20,-24,30,-20,32,-32,24,-32,30,-28,36,-40,24,-40,42,
%U A138557 -24,42,-48,35,-44,46,-32,43,-50,32,-48,52,-40,60,-48,40,-60,60,-40
%N A138557 Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^10)^7 / (eta(q^2)^3 * eta(q^5)^2 * eta(q^20)^2) in powers of q.
%C A138557 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138557 G. C. Greubel, <a href="/A138557/b138557.txt">Table of n, a(n) for n = 1..1000</a>
%H A138557 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. 337, Eq. (3.19).
%H A138557 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138557 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138557 Expansion of q * (f(q) / chi(q)^3) * (f(q^5)^3 / chi(q^5)) in powers of q where chi(), f() are Ramanujan theta functions.
%F A138557 Expansion of q * f(q^5)^5 / f(q) - q^2 * f(-q^10)^5 / f(-q^2) in powers of q where f() is a Ramanujan theta function.
%F A138557 Euler transform of period 20 sequence [ -2, 1, -2, -1, 0, 1, -2, -1, -2, -4, -2, -1, -2, 1, 0, -1, -2, 1, -2, -4, ...].
%F A138557 a(n) is multiplicative with a(2^e) = -2^e if e>0, a(5^e) = 5^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).
%F A138557 G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 80^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138558.
%F A138557 G.f.: Sum_{k>0} -(-1)^k * k * x^k * (1 - x^(2*k)) * (1 - x^(6*k)) / (1 - x^(10*k)).
%F A138557 G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(2*k-1)) * (1 + x^(2*k))^2 * (1 + x^(10*k-5))^2 * (1 - x^(10*k))^3.
%F A138557 G.f.: Sum_{k>0} f(10*k-1) - f(10*k-3) - f(10*k-7) + f(10*k-9) where f(k) := x^k / (1 + x^k)^2.
%F A138557 a(n) = -(-1)^n * A129303(n).
%e A138557 G.f. = q - 2*q^2 + 2*q^3 - 4*q^4 + 5*q^5 - 4*q^6 + 6*q^7 - 8*q^8 + 7*q^9 + ...
%t A138557 a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, n/# KroneckerSymbol[ 20, #] &]]; (* _Michael Somos_, Sep 08 2015 *)
%t A138557 a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^5]^5 / QPochhammer[ -q] - q^2 QPochhammer[ q^10]^5 / QPochhammer[ q^2], {q, 0, n}]; (* _Michael Somos_, Sep 08 2015 *)
%t A138557 a[ n_] := SeriesCoefficient[ q QPochhammer[ -q] QPochhammer[ q, -q]^3 QPochhammer[ -q^5] ^3 QPochhammer[ q^5, -q^5], {q, 0, n}]; (* _Michael Somos_, Sep 08 2015 *)
%o A138557 (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, n/d * kronecker(20, d)))};
%o A138557 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^7 / (eta(x^2 + A)^3 * eta(x^5 + A)^2 * eta(x^20 + A)^2), n))};
%Y A138557 Cf. A129303, 138558.
%K A138557 sign,mult
%O A138557 1,2
%A A138557 _Michael Somos_, Mar 24 2008