OFFSET
-1,9
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 1/q * chi(q) * chi(q^23) in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q^2) * eta(q^46))^2 / (eta(q) * eta(q^4) * eta(q^23) * eta(q^92)) in powers of q.
Euler transform of a period 92 sequence.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v - 1)^2 - u*v * (2 - 2*v + v^2 - u) * (2 - 2*u + u^2 - v) / 2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (92 t)) = f(t) where q = exp(2 Pi i t).
G.f.: 1/x * Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(46*k - 23)).
a(n) ~ exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017
EXAMPLE
G.f. = 1/q + 1 + q^2 + q^3 + q^4 + q^5 + q^6 + 2*q^7 + 2*q^8 + 2*q^9 + 2*q^10 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ -q, q^2] QPochhammer[ -q^23, q^46], {q, 0, n}];
a[ n_] := SeriesCoefficient[ 1/q Product[ 1 + q^k, {k, 1, n + 1, 2}] Product[ 1 + q^k, {k, 23, n + 1, 46}], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^46 + A))^2 / (eta(x + A) * eta(x^4 + A) * eta(x^23 + A) * eta(x^92 + A)), n))};
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( prod( k=1, n, 1 + k%2 * x^k, 1 + A) * prod( k=1, n\23, 1 + k%2 * x^(23*k), 1 + A), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, May 21 2013
STATUS
approved