A392023 Expansion of the series e(q)/e(-q), where the series e(q) is Morier-Genoud and Ovsienko's q-analog of the real number e = exp(1).

1, 2, 2, 4, 6, 6, 8, 2, -6, -26, -62, -96, -156, -150, -106, 150, 698, 1590, 3142, 4630, 6234, 5436, 528, -12876, -40696, -83918, -148010, -210998, -248122, -175238, 126894, 823188, 2138058, 4125838, 6718458, 9027984, 9272758, 3875732, -12847246, -47830012, -108634298, -195967684, -297456200, -368423014, -314077440

Offset

0,2

Comments

Notice the unusual pattern in the signs of the coefficients of the series e(q)/e(-q), which appears to consist of a group of either 7 or 8 consecutive positive coefficients followed by a group of either 7 or 8 consecutive negative coefficients. Empirically, sign(a(n)) - sign(a(n+15)) is frequently zero. Similar behavior is seen in the coefficients of the power series e(-q) (A392020), 1/e(-q) (A392021) and e(q)*e(-q) (A392022).

Links

Table of n, a(n) for n=0..44.

Formula

e(q)/e(-q) = 1 + 2*q + 2*q^2 + 4*q^3 + 6*q^4 + 6*q^5 + 8*q^6 + 2*q^7 - 6*q^8 - 26*q^9 - 62*q^10 - 96*q^11 - 156*q^12 - 150*q^13 - 106*q^14 + 150*q^15 + 698*q^16 + 1590*q^17 + 3142*q^18 + 4630*q^19 + 6234*q^20 + 5436*q^21 + 528*q^22 - ...

Maple

# uses 2 X 2 matrices to determine rational function approximations to e(q)/e(-q) - see  Morier-Genoud and Ovsienko, Section 2.4
# calculate the partial quotients of the simple continued fraction expansion of e
c := proc(n) if n = 1 then 2 elif irem(n, 3) = 0 then (2/3)*n else 1 end if end proc:
# define the q-analog of the integer n
f(n, q) := simplify( (q^n - 1)/(q - 1) ):
with(LinearAlgebra):
Mat(n, q) := Matrix( [[f(c(2*n-1), q), q^(c(2*n-1))], [1, 0]] ).Matrix([[q*f(c(2*n), q), 1], [q^(c(2 n)), 0]] ):
Mat_prod := proc(n, q) option remember; if n = 1 then Mat(1, q) else simplify( Mat_prod(n-1, q).Mat(n, q) ) end if end proc:
# series expansion of e(q)/e(-q)
S := series( simplify( Mat_prod(14, q)[1, 1])/simplify(Mat_prod(14, q)[2, 1]) * simplify(Mat_prod(14, -q)[2, 1])/simplify(Mat_prod(14, -q)[1, 1]), q, 101 ):
seq(coeftayl(S, q = 0, n), n = 0..100);

Crossrefs

Cf. A001113, A003417, A337589, A392020 (e(-q)), A392021 (1/e(-q)), A392022 (e(q)*e(-q)).

Sequence in context: A160731 A323647 A302928 * A098214 A178883 A109832

Adjacent sequences: A392020 A392021 A392022 * A392024 A392025 A392026

Keyword

sign,easy

Author

Peter Bala, Jan 06 2026

Status

approved