A109064 Expansion of eta(q)^5 / eta(q^5) in powers of q.

1, -5, 5, 10, -15, -5, -10, 30, 25, -35, 5, -60, 30, 60, -30, 10, -55, 80, 35, -100, -15, -60, 60, 110, -50, -5, -60, 100, 90, -150, -10, -160, 105, 120, -80, 30, -105, 180, 100, -120, 25, -210, 60, 210, -180, -35, -110, 230, 110, -215, 5, -160, 180, 260

Offset

0,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 151.

Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, 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]

Formula

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

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.

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

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

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

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

a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A285896(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

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

Example

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 + ...

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d, 5)=0, -4, -5), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 07 2017

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^5 / QPochhammer[ q^5], {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -5 DivisorSum[ n, # KroneckerSymbol[ 5, #] &]]; (* Michael Somos, May 19 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^5 + A), n))};
(PARI)  {a(n) = if( n<1, n==0, -5 * sumdiv(n, d, d * kronecker(5, d)))}; /* Michael Somos, May 19 2015 */
(Magma) A := Basis( ModularForms( Gamma1(5), 2), 54); A[1] - 5*A[2] + 5*A[3]; /* Michael Somos, May 19 2015 */

Crossrefs

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

Sequence in context: A112436 A309457 A285932 * A138506 A000728 A242895

Adjacent sequences: A109061 A109062 A109063 * A109065 A109066 A109067

Keyword

sign

Author

Michael Somos, Jun 17 2005

Status

approved