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Expansion of eta(q)^5 / eta(q^5) in powers of q.
12

%I #43 Jan 29 2024 01:20:09

%S 1,-5,5,10,-15,-5,-10,30,25,-35,5,-60,30,60,-30,10,-55,80,35,-100,-15,

%T -60,60,110,-50,-5,-60,100,90,-150,-10,-160,105,120,-80,30,-105,180,

%U 100,-120,25,-210,60,210,-180,-35,-110,230,110,-215,5,-160,180,260

%N Expansion of eta(q)^5 / eta(q^5) in powers of q.

%C Number 12 of the 74 eta-quotients listed in Table I of Martin (1996).

%H Seiichi Manyama, <a href="/A109064/b109064.txt">Table of n, a(n) for n = 0..10000</a>

%H W. Duke, <a href="http://dx.doi.org/10.1090/S0273-0979-05-01047-5">Continued fractions and modular functions</a>, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 151.

%H Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.

%H G. N. Watson, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002174499&amp;physid=PHYS_0110">Ramanujans Vermutung über Zerfällungsanzahlen</a>, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See the expression B^5/C in the notation of p. 106. [Added by _N. J. A. Sloane_, Nov 13 2009]

%F Euler transform of period 5 sequence [ -5, -5, -5, -5, -4, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + 2 * u*v*w + u^2*w - 4 * u*w^2.

%F a(n) = -5 * b(n) where b() is multiplicative with a(0) = 1, b(p^e) = 1 if p=5, b(p^e) = b(p) * b(p^(e-1)) - Kronecker(5, p) * p * b(p^(e-2)) otherwise. - _Michael Somos_, May 19 2015

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 5^(5/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053723. - _Michael Somos_, May 19 2015

%F G.f.: Product_{k>0} (1 - x^k)^5 / (1 - x^(5*k)).

%F a(n) = -5 * A109091(n), unless n=0. a(n) = (-1)^n * A138506(n). a(5*n) = a(n).

%F a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 29 2017

%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(3*sqrt(5)) = 1.471273... . - _Amiram Eldar_, Jan 29 2024

%e G.f. = 1 - 5*q + 5*q^2 + 10*q^3 - 15*q^4 - 5*q^5 - 10*q^6 + 30*q^7 + 25*q^8 + ...

%p with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

%p `if`(irem(d, 5)=0, -4, -5), d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..70); # _Alois P. Heinz_, Jan 07 2017

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^5 / QPochhammer[ q^5], {q, 0, n}]; (* _Michael Somos_, May 19 2015 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], -5 DivisorSum[ n, # KroneckerSymbol[ 5, #] &]]; (* _Michael Somos_, May 19 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^5 + A), n))};

%o (PARI) {a(n) = if( n<1, n==0, -5 * sumdiv(n, d, d * kronecker(5, d)))}; /* _Michael Somos_, May 19 2015 */

%o (Magma) A := Basis( ModularForms( Gamma1(5), 2), 54); A[1] - 5*A[2] + 5*A[3]; /* _Michael Somos_, May 19 2015 */

%Y Cf. A053723, A109091, A138506, A277212, A285896.

%K sign

%O 0,2

%A _Michael Somos_, Jun 17 2005