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A143323 Expansion of eta(q^2)^4 * eta(q^5) * eta(q^20)^2 / ( eta(q) * eta(q^4)^2 * eta(q^10)^2 ) in powers of q.

%I A143323 #19 Feb 16 2025 08:33:08
%S A143323 1,1,-2,-1,1,-2,-2,1,3,1,0,2,0,-2,-2,-1,0,3,0,-1,4,0,-2,-2,1,0,-4,2,2,
%T A143323 -2,0,1,0,0,-2,-3,0,0,0,1,2,4,-2,0,3,-2,-2,2,3,1,0,0,0,-4,0,-2,0,2,0,
%U A143323 2,2,0,-6,-1,0,0,-2,0,4,-2,0,3,0,0,-2,0,0,0,0,-1,5,2,-2,-4,0,-2,-4,0,2,3,0,2,0,-2,0,-2,0,3,0,-1,2,0,-2,0,4
%N A143323 Expansion of eta(q^2)^4 * eta(q^5) * eta(q^20)^2 / ( eta(q) * eta(q^4)^2 * eta(q^10)^2 ) in powers of q.
%C A143323 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A143323 G. C. Greubel, <a href="/A143323/b143323.txt">Table of n, a(n) for n = 1..1000</a>
%H A143323 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. 338, Eq. (3.22), p. 342, Eq. (3.41).
%H A143323 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143323 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143323 Expansion of q * phi(-q^2) * chi(q) * psi(q^10) * chi(-q^5) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%F A143323 Expansion of q * phi(-q^2)^2 * psi(-q^5)^2 / (f(-q) * f(-q^5)) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Apr 07 2015
%F A143323 Euler transform of period 20 sequence [ 1, -3, 1, -1, 0, -3, 1, -1, 1, -2, 1, -1, 1, -3, 0, -1, 1, -3, 1, -2, ...].
%F A143323 Multiplicative with a(2^e) = -(-1)^e unless e=0, a(p^e) = 1 if p=5, a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e+1 if p == 1, 9 (mod 20), a(p^e) = (e+1)(-1)^e if p == 3, 7 (mod 20).
%F A143323 G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A129391.
%F A143323 G.f.: Sum_{k>0} -(-1)^k F(x^(2*k - 1)) where F(x) = x * (1 + x) * (1 - x^2) / (1 + x^5).
%F A143323 G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2 * (1 - x^(5*k)) * ( 1 + x^(10*k))^2.
%F A143323 a(n) = -(-1)^n * A111949(n).
%e A143323 G.f. = q + q^2 - 2*q^3 - q^4 + q^5 - 2*q^6 - 2*q^7 + q^8 + 3*q^9 + q^10 + ...
%t A143323 a[ n_] := SeriesCoefficient[ 2 q^(13/8) EllipticTheta[ 4, 0, q^2]^2 QPochhammer[ q^20]^2 / ( QPochhammer[ q] EllipticTheta[ 2, 0, q^(5/2)]), {q, 0, n}]; (* _Michael Somos_, Apr 07 2015 *)
%o A143323 (PARI) {a(n) = if( n<1, 0, (-1)^n * (qfrep([2, 1; 1, 3], n)[n] - qfrep([1, 0; 0, 5], n)[n] ))};
%o A143323 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^5 + A) * eta(x^20 + A)^2 / ( eta(x + A) * eta(x^4 + A)^2 * eta(x^10 + A)^2 ), n))};
%Y A143323 Cf. A111949, A129391.
%K A143323 sign,mult
%O A143323 1,3
%A A143323 _Michael Somos_, Aug 07 2008