A145724 - OEIS

%I #10 Mar 12 2021 22:24:45

%S 1,-1,2,-3,5,-6,10,-13,19,-25,36,-46,64,-82,110,-139,184,-231,300,

%T -375,480,-596,754,-930,1165,-1428,1772,-2162,2662,-3230,3952,-4773,

%U 5800,-6976,8430,-10093,12136,-14476,17320,-20585,24526,-29044,34466,-40684,48095

%N Expansion of q * f(-q^20) / (f(q) * chi(-q^5)) in powers of q where f(), chi() are Ramanujan theta functions.

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

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

%F Euler transform of period 20 sequence [ -1, 2, -1, 1, 0, 2, -1, 1, -1, 2, -1, 1, -1, 2, 0, 1, -1, 2, -1, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A147701.

%F Convolution inverse of A145723.

%F a(n) ~ -(-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n/5)) /  (2^(5/2) * 5^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018

%e G.f. = q - q^2 + 2*q^3 - 3*q^4 + 5*q^5 - 6*q^6 + 10*q^7 - 13*q^8 + 19*q^9 + ...

%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q^20] QPochhammer[ -q^5, q^5] / QPochhammer[ -q], {q, 0, n}]; (* _Michael Somos_, Sep 05 2015 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A) * eta(x^20 + A) / (eta(x^2 + A)^3 * eta(x^5 + A)), n))};

%Y Cf. A145723, A147701.

%K sign

%O 1,3

%A _Michael Somos_, Nov 10 2008