A109091 Expansion of (1 - eta(q)^5 / eta(q^5)) / 5 in powers of q.

1, -1, -2, 3, 1, 2, -6, -5, 7, -1, 12, -6, -12, 6, -2, 11, -16, -7, 20, 3, 12, -12, -22, 10, 1, 12, -20, -18, 30, 2, 32, -21, -24, 16, -6, 21, -36, -20, 24, -5, 42, -12, -42, 36, 7, 22, -46, -22, 43, -1, 32, -36, -52, 20, 12, 30, -40, -30, 60, -6, 62, -32, -42, 43, -12, 24, -66, -48, 44, 6, 72, -35, -72, 36, -2, 60, -72

Offset

1,3

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. This is the expression B^5/C in the notation of p. 106. [Added by N. J. A. Sloane, Nov 13 2009]

Formula

G.f.: (1 - Product_{k>0} (1 - x^k)^5 / (1 - x^(5*k))) / 5 = Sum_{k>0} x^k * (1 - x^k)^2 * (1 + x^(6*k) - 4*x^(2*k) * (1 + x^k +x^(2*k))) / (1 - x^(5*k))^2.

-5*a(n) = A109064(n) unless n = 0.

a(n) = A284097(n) + A284103(n) - A284280(n) - A284281(n) = A284150(n) - A284152(n). - Seiichi Manyama, Apr 01 2017

L.g.f.: -log(1/(1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...))))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(15*sqrt(5)) = 0.294254... . - Amiram Eldar, Jan 29 2024

Example

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

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ d KroneckerSymbol[ 5, d], {d, Divisors@n}]]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^5 / QPochhammer[ q^5]) / 5, {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, A = x * O(x^n); -1/5 * polcoeff( eta(x + A)^5 / eta(x^5 + A), n))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d * kronecker(5, d)))} /* Michael Somos, Mar 21 2008 */
(Ruby)
def s(k, m, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == m}
  s
end
def A109091(n)
  (1..n).map{|i| s(5, 1, i) + s(5, 4, i) - s(5, 2, i) - s(5, 3, i)}
end # Seiichi Manyama, Apr 01 2017

Crossrefs

Cf. A003823, A007325, A078905, A109064, A229793.

Cf. A284097, A284103, A284150, A284152, A284280, A284281.

Sequence in context: A131108 A128255 A154948 * A138507 A209579 A205699

Adjacent sequences: A109088 A109089 A109090 * A109092 A109093 A109094

Keyword

sign,mult

Author

Michael Somos, Jun 18 2005

Status

approved