A067630 Denominators in power series for cos(x)*cosh(x).
1, 6, 2520, 7484400, 81729648000, 2375880867360000, 151476660579404160000, 18608907752179801056000000, 4015057936610313875842560000000, 1419041926536183233139035980800000000, 778117449996850714059458989711872000000000
Offset: 0
Keywords
Programs
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Maple
f:= gfun:-rectoproc({a(n) - (64*n^4-96*n^3+44*n^2-6*n)*a(n-1), a(0)=1}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Aug 17 2021 -
Mathematica
a[n_] := (4*n)!/4^n; Array[a, 10, 0] (* Amiram Eldar, Jan 18 2021 *)
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PARI
my(x='x+O('x^50), v=apply(denominator, Vec(cos(x)*cosh(x)))); vector(#v\4, k, v[4*k-3]) \\ Michel Marcus, Jan 18 2021
Formula
cos(x)*cosh(x) = Sum_{n>=0} (-1)^n*x^(4*n)/a(n).
a(n) = (4*n)! / 4^n = A000680(2*n).
E.g.f.: 1/(1-x^4/4). - Mohammad K. Azarian, Mar 20 2012
a(n) = n!*A060706(n). - Bruno Berselli, Mar 21 2012
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=0} 1/a(n) = (cos(sqrt(2)) + cosh(sqrt(2)))/2.
Sum_{n>=0} (-1)^n/a(n) = cos(1)*cosh(1). (End)
D-finite with recurrence: a(n) - (64*n^4 - 96*n^3 + 44*n^2 - 6*n)*a(n-1) = 0. - Georg Fischer, Aug 17 2021