A117410 - OEIS

%I #22 Nov 18 2022 08:56:15

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

%N Expansion of q^(-5/24) * eta(q^2)^3 / eta(q) 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="/A117410/b117410.txt">Table of n, a(n) for n = 0..1000</a>

%H B. Gordon and D. Sinor, <a href="https://doi.org/10.1007/BFb0086404">Multiplicative properties of eta-products</a>, Number theory, Madras 1987, pp. 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989. see page 183. MR1019331 (90k:11050).

%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 psi(x)^2 * chi(-x) = f(-x)^2 / chi(-x)^3 = f(-x)^5 / phi(-x)^3 = f(-x^2)^2 / chi(-x) = f(-x^2)^3 / f(-x) = psi(x) * f(-x^2) = f(x) * f(-x^4) = phi(-x)^2 / chi(-x)^5 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Jan 31 2015

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

%F Given A = A0 + A1 + A2 + A3 + A4 is the 5-section, then 0 = A3 * A1^2 - A2 * A4^2.

%F Given A = A0 + A1 + A2 + A3 + A4 + A5 + A6 is the 7-section, then 0 = A0*A6 + A1*A5 + A2*A4 + 4*A3^2, A3 = x^10 * A(x^49).

%F G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2.

%F A107034(n) = (-1)^n * a(n).

%e G.f. = 1 + x - x^2 - x^4 - 2*x^5 + x^6 - x^7 - x^8 + x^10 + x^11 - x^12 + x^13 + ...

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

%p # Uses EulerTransform from A358369.

%p a := EulerTransform(BinaryRecurrenceSequence(0, 1, -2)):

%p seq(a(n), n = 0..104); # _Peter Luschny_, Nov 17 2022

%t a[ n_] := SeriesCoefficient[ QPochhammer[x^2]^3 / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Jan 31 2015 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A), n))};

%o (PARI) q='q+O('q^99); Vec(eta(q^2)^3/eta(q)) \\ _Altug Alkan_, Apr 17 2018

%o (Sage) # uses[EulerTransform from A166861]

%o b = BinaryRecurrenceSequence(0, 1, -2)

%o a = EulerTransform(b)

%o print([a(n) for n in range(105)]) # _Peter Luschny_, Nov 17 2022

%Y Cf. A107034.

%K sign

%O 0,6

%A _Michael Somos_, Mar 13 2006