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

%I A109091 #40 Aug 11 2025 06:24:02
%S A109091 1,-1,-2,3,1,2,-6,-5,7,-1,12,-6,-12,6,-2,11,-16,-7,20,3,12,-12,-22,10,
%T A109091 1,12,-20,-18,30,2,32,-21,-24,16,-6,21,-36,-20,24,-5,42,-12,-42,36,7,
%U A109091 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
%N A109091 Expansion of (1 - eta(q)^5 / eta(q^5)) / 5 in powers of q.
%H A109091 Seiichi Manyama, <a href="/A109091/b109091.txt">Table of n, a(n) for n = 1..10000</a>
%H A109091 G. N. Watson, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0179">Ramanujans Vermutung über Zerfällungsanzahlen</a>, 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]
%F A109091 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.
%F A109091 -5*a(n) = A109064(n) unless n = 0.
%F A109091 a(n) = A284097(n) + A284103(n) - A284280(n) - A284281(n) = A284150(n) - A284152(n). - _Seiichi Manyama_, Apr 01 2017
%F A109091 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
%F A109091 Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(15*sqrt(5)) = 0.294254... . - _Amiram Eldar_, Jan 29 2024
%e A109091 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 + ...
%t A109091 a[ n_] := If[ n < 1, 0, Sum[ d KroneckerSymbol[ 5, d], {d, Divisors@n}]]; (* _Michael Somos_, Apr 26 2015 *)
%t A109091 a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^5 / QPochhammer[ q^5]) / 5, {q, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%o A109091 (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))};
%o A109091 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d * kronecker(5, d)))} /* _Michael Somos_, Mar 21 2008 */
%o A109091 (Ruby)
%o A109091 def s(k, m, n)
%o A109091   s = 0
%o A109091   (1..n).each{|i| s += i if n % i == 0 && i % k == m}
%o A109091   s
%o A109091 end
%o A109091 def A109091(n)
%o A109091   (1..n).map{|i| s(5, 1, i) + s(5, 4, i) - s(5, 2, i) - s(5, 3, i)}
%o A109091 end # _Seiichi Manyama_, Apr 01 2017
%Y A109091 Cf. A003823, A007325, A078905, A109064, A229793.
%Y A109091 Cf. A284097, A284103, A284150, A284152, A284280, A284281.
%K A109091 sign,mult
%O A109091 1,3
%A A109091 _Michael Somos_, Jun 18 2005