OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..100
FORMULA
a(n)/[n]_q! is the coefficient of x^n in the expansion of exp_q(x)*cosh_q(x) when q->2, and cosh_q(x) = Sum_{n>=0} x^(2n)/[2n]_q!, and exp_q(x) is the q-exponential function, and [n]_q! is the q-factorial of n.
From Vaclav Kotesovec, Jun 11 2025: (Start)
a(n) ~ c * 2^(n^2/4), where
c = 3.89569562162... if mod(n,4) = 0,
c = 3.68597474538... if mod(n,4) = 1 or 3,
c = 3.47627317983... if mod(n,4) = 2. (End)
MATHEMATICA
nn = 22; eq[z_] := Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];
coshq[z_] := Sum[z^(2 n)/FunctionExpand[QFactorial[(2 n), q]], {n, 0, nn}];
Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}]*
CoefficientList[Series[coshq[z]*eq[z] /. q -> 2, {z, 0, nn}], z]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Jul 14 2017
STATUS
approved