A113260 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.

1, -3, -2, 5, 1, 6, -6, -11, 7, -3, 12, -10, -12, 18, -2, 21, -16, -21, 20, 5, 12, -36, -22, 22, 1, 36, -20, -30, 30, 6, 32, -43, -24, 48, -6, 35, -36, -60, 24, -11, 42, -36, -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

Offset

1,2

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=1..74.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

Formula

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).

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

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

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A197293 A099643 A224344 * A051543 A265574 A129538

Adjacent sequences: A113257 A113258 A113259 * A113261 A113262 A113263

Keyword

sign,mult

Author

Michael Somos, Oct 20 2005

Status

approved