OFFSET
0,3
COMMENTS
REFERENCES
Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 253 Eq. (8.12)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Shaun Cooper, On Ramanujan's function k(q)=r(q)r^2(q^2), Ramanujan J., 20 (2009), 311-328; see p. 312, eq. (1.4).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of (5 * phi(-q) * phi(-q^5)^3 - phi(-q)^3 * phi(-q^5)) / 4 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 10 sequence [-1, -3, -1, -3, -4, -3, -1, -3, -1, -4, ...].
a(n) = -b(n) where b() is multiplicative with b(5^e) = 1, b(2^e) = 2 - ((-2)^(e+1) - 1) / (-2 - 1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 7 (mod 10).
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k))^2 * (1 - x^(5*k)) / (1 + x^(5*k))^2.
G.f.: 1 + Sum_{k>0} (-1)^k * k * x^k / (1 - x^k) * Kronecker(5, k).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 2000^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A129303.
a(n) = (-1)^n * A113185(n).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(12*sqrt(5)) = 0.367818... . - Amiram Eldar, Jan 28 2024
EXAMPLE
G.f. = 1 - q - 3*q^2 + 2*q^3 + q^4 - q^5 + 6*q^6 + 6*q^7 - 7*q^8 - 7*q^9 +...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ 5, #] # (-1)^# &]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^2]^2 QPochhammer[ q^5]^3 / QPochhammer[ q^10]^2, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ (5 EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^5]^3 - EllipticTheta[ 4, 0, q]^3 EllipticTheta[ 4, 0, q^5])/4, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
PROG
(PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker(5, d) * d * (-1)^d))};
(PARI) {a(n) = my(A, p, e, a1); if( n<1, n==0, A = factor(n); -prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p>2, p *= kronecker(5, p); (p^(e+1) - 1) / (p - 1), (5 + (-2)^(e+1)) / 3)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^2 * eta(x^5 + A)^3 / eta(x^10 + A)^2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 08 2007, Mar 20 2008
STATUS
approved