A113260 - OEIS

%I #20 Jan 22 2024 01:24:28

%S 1,-3,-2,5,1,6,-6,-11,7,-3,12,-10,-12,18,-2,21,-16,-21,20,5,12,-36,

%T -22,22,1,36,-20,-30,30,6,32,-43,-24,48,-6,35,-36,-60,24,-11,42,-36,

%U -42,60,7,66,-46,-42,43,-3,32,-60,-52,60,12,66,-40,-90,60,-10,62,-96,-42,85,-12,72,-66,-80,44,18,72,-77,-72,108

%N Expansion of (-1 + psi(q)^5/psi(q^5) - 25q^2 psi(q)*psi(q^5)^3)/5 in powers of q where psi(q) is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 249, Entry 8(iv).

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.

%F a(n) is multiplicative with a(2^e) = ((-2)^(e+2)-1)/3, a(5^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 4 (mod 5), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 2, 3 (mod 5).

%F G.f.: Sum_{k>0} (k*x^k / (1 + x^k)) * Kronecker(5, k).

%F A113259(n) = 5*a(n) if n > 0.

%F Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(10*sqrt(5)) = 0.441382... . - _Amiram Eldar_, Jan 22 2024

%e G.f. = x - 3*x^2 - 2*x^3 + 5*x^4 + x^5 + 6*x^6 - 6*x^7 - 11*x^8 + 7*x^9 + ... - _Michael Somos_, Sep 07 2018

%t a[ n_] := If[ n < 1, 0, -DivisorSum[ n, # KroneckerSymbol[ 5, #] (-1)^(n/#) &]]; (* _Michael Somos_, Sep 07 2018 *)

%o (PARI) {a(n) = if( n<1, 0, -sumdiv(n, d, d * kronecker(5, d) * (-1)^(n/d)))};

%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p==2, ((-2)^(e+2) - 1)/3, p *= kronecker(5, p); (p^(e+1) - 1) / (p - 1))))};

%Y Cf. A000122, A000700, A010054, A113259, A121373.

%K sign,mult

%O 1,2

%A _Michael Somos_, Oct 20 2005