A107034 - OEIS

%I #11 Mar 12 2021 22:24:43

%S 1,-1,-1,0,-1,2,1,1,-1,0,1,-1,-1,-1,0,-2,1,0,0,1,2,-1,0,1,0,1,0,1,1,

%T -1,-3,0,-1,1,-1,-1,0,0,0,1,-2,0,1,0,1,0,1,0,0,1,2,1,0,-1,1,-3,0,1,0,

%U -1,-1,0,1,0,0,-2,0,-1,-1,0,-2,1,1,0,0,1,0,0,1

%N Expansion of f(-x) * f(-x^4) in powers of x where f() is a Ramanujan theta function.

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

%D H. Kahl, G. Koehler, Components of Hecke theta series, J. Math. Anal. Appl. 232 (1999), no. 2, 312-331, see page 320. MR1683136 (2000e:11051)

%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 f(x)^2 / chi(x)^3 = f(x)^5 / phi(x)^3 = f(-x^2)^2 / chi(x) = f(-x^2) * psi(-x) = f(-x^2)^3 / f(x) = phi(x)^2 / chi(x)^5 = psi(-x)^2 * chi(x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Jan 29 2015

%F Expansion of q^(-5/24) * eta(q) * eta(q^4) in powers of q.

%F Euler transform of period 4 sequence [-1, -1, -1, -2, ...].

%F G.f. Product_{k>0} (1 - x^k) * (1 - x^(4*k)).

%e G.f. = 1 - x - x^2 - x^4 + 2*x^5 + x^6 + x^7 - x^8 + x^10 - x^11 - x^12 - ...

%e G.f. = q^5 - q^29 - q^53 - q^101 + 2*q^125 + q^149 + q^173 - q^197 + q^245 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4], {x, 0, n}]; (* _Michael Somos_, Jan 29 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A), n))};

%K sign

%O 0,6

%A _Michael Somos_, May 09 2005