OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
L.-C. Shen, On the additive formulas of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5, Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345. See p. 343, Eq. (3.44).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of q * (f(q)^3 / chi(q)) * (f(q^5) / chi(q^5)^3) in powers of q where chi(), f() are Ramanujan theta functions.
Expansion of ( f(q)^5 / f(q^5) - f(-q^2)^5 / f(-q^10) ) / 5 in powers of q where f() is a Ramanujan theta function.
Euler transform of period 20 sequence [ 2, -5, 2, -3, 0, -5, 2, -3, 2, -4, 2, -3, 2, -5, 0, -3, 2, -5, 2, -4, ...].
a(n) is multiplicative with a(2^e) = -(-2)^e if e>0, 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. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138557.
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k))^3 * (1 - x^(5*k)) * (1 - x^(10*k-5)) * (1 + x^(10*k))^2.
G.f.: Sum_{k>0} (-1)^k * ( f(5*k-1) + f(5*k-2) - f(5*k-3) - f(5*k-4) ) where f(k) := k * x^k / (1 - x^(2*k)).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . - Amiram Eldar, Feb 01 2024
EXAMPLE
G.f. = q + 2*q^2 - 2*q^3 - 4*q^4 + q^5 - 4*q^6 - 6*q^7 + 8*q^8 + 7*q^9 + 2*q^10 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, # Mod[n/#, 2] KroneckerSymbol[ 5, #] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q]^5 / QPochhammer[ -q^5] - QPochhammer[ q^2]^5 / QPochhammer[ q^10]) / 5, {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q]^3 Pochhammer[ q, -q] QPochhammer[ -q^5] QPochhammer[ q^5, -q^5]^3, {q, 0, n}]; (* Michael Somos, Sep 08 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, (n/d%2) * d * kronecker(5, d)))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^5 + A)^2 * eta(x^20 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^3), n))};
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Michael Somos, Mar 24 2008, Mar 25 2008
STATUS
approved