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A138507 Expansion of (f(q)^5 / f(q^5) - 1) / 5 in powers of q where f() is a Ramanujan theta function.

%I A138507 #15 Feb 16 2025 08:33:08
%S A138507 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 A138507 1,-12,-20,18,30,-2,32,21,-24,-16,-6,-21,-36,20,24,5,42,12,-42,-36,7,
%U A138507 -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,-72,24
%N A138507 Expansion of (f(q)^5 / f(q^5) - 1) / 5 in powers of q where f() is a Ramanujan theta function.
%C A138507 Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
%H A138507 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A138507 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A138507 a(n) is multiplicative with a(2^e) = ((-2)^(e+1) - 1) / 3, a(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 7 (mod 10), a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10).
%F A138507 G.f.: (Product_{k>0} (1 - (-x)^k)^5 / (1 - (-x)^(5*k)) - 1) / 5.
%F A138507 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 A138507 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 A138507 q + q^2 - 2*q^3 - 3*q^4 + q^5 - 2*q^6 - 6*q^7 + 5*q^8 + 7*q^9 + ...
%o A138507 (PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, d * kronecker(5, d)))}
%o A138507 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(-x + A)^5 / eta(-x^5 + A) - 1) / 5, n))}
%Y A138507 -(-1)^n * A109091(n) = a(n). A138506(n) = 5 * a(n) unless n=0.
%Y A138507 Cf. A000122, A000700, A010054, A121373.
%K A138507 sign,mult
%O A138507 1,3
%A A138507 _Michael Somos_, Mar 21 2008