OFFSET
-1,13
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = -1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1) * chi(-q) * chi(-q^39) / (chi(-q^3) * chi(-q^13)) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q) * eta(q^6) * eta(q^26) * eta(q^39) / (eta(q^2) * eta(q^3) * eta(q^13) * eta(q^78)) in powers of q.
Euler transform of a period 78 sequence.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (v - u^2) * (w^2 - v) - 2*u*w * (1 + v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (78 t)) = 1 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A262950.
G.f.: (1/x) * (Product_{k>0} P(x^k))^-1 where P(x) is the 78th cyclotomic polynomial of degree 24.
a(n) = A058755(n) unless n = 0.
Convolution inverse is A262950.
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/39)) / (2 * 39^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2018
EXAMPLE
G.f. = 1/q - 1 - q^4 + q^5 - q^6 + q^7 - q^10 + 2*q^11 - q^12 + q^15 + ...
MATHEMATICA
QP = QPochhammer; s = QP[q]*QP[q^6]*QP[q^26]*(QP[q^39]/(QP[q^2]*QP[q^3]* QP[q^13]*QP[q^78])) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^26 + A) * eta(x^39 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^13 + A) * eta(x^78 + A)), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 06 2007
STATUS
approved