A113259 Expansion of psi(x)^5 / psi(x^5) - 25*x^2 * psi(x) * psi(x^5)^3 in powers of x where psi() is a Ramanujan theta function.

1, 5, -15, -10, 25, 5, 30, -30, -55, 35, -15, 60, -50, -60, 90, -10, 105, -80, -105, 100, 25, 60, -180, -110, 110, 5, 180, -100, -150, 150, 30, 160, -215, -120, 240, -30, 175, -180, -300, 120, -55, 210, -180, -210, 300, 35, 330, -230, -210, 215, -15, 160, -300, -260, 300, 60, 330, -200, -450

Offset

0,2

Comments

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

References

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

Links

Table of n, a(n) for n=0..58.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

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

a(n) = 5 * A113260(n) if n>0.

Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(2*sqrt(5)) = 2.2069106... . - Amiram Eldar, Jan 28 2024

Example

G.f. = 1 + 5*x - 15*x^2 - 10*x^3 + 25*x^4 + 5*x^5 + 30*x^6 - 30*x^7 - 55*x^8 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], -5 DivisorSum[ n, # KroneckerSymbol[ 5, #] (-1)^(n/#) &]]; (* Michael Somos, Sep 07 2018 *)

Prog

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

Crossrefs

Cf. A113260.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A290829 A290837 A302841 * A298265 A366419 A309620

Adjacent sequences: A113256 A113257 A113258 * A113260 A113261 A113262

Keyword

sign

Author

Michael Somos, Oct 20 2005

Status

approved