OFFSET
0,2
COMMENTS
Apart from a(0) = 1, this is the sequence of denominators in the convergents of the simple continued fraction expansion of (e - 1)/(e + 1). See A086403 for the numerators. - Peter Bala, Dec 14 2025
LINKS
Robert Israel, Table of n, a(n) for n = 0..365
FORMULA
a(n)/|A002119(n)| tends to 1.8591409... = (e+1)/2.
a(n)/A001517(n) tends to 0.68393972... = 2e/(e+1).
E.g.f.: cosh((1-sqrt(1-4*x))/2)/sqrt(1-4*x). - Vladimir Kruchinin, May 03 2016
a(n) = Sum_{k=0..n/2}((2*n-2*k)!/((n-2*k)!*(2*k)!)). - Vladimir Kruchinin, May 03 2016
a(n) = ((-1)^n*sqrt(Pi*exp(-1))*BesselI((2*n+1)/2, 1/2))/2 + (BesselK((2*n+1)/2, 1/2)*cosh(1/2))/sqrt(Pi), where BesselI(n,x) is the modified Bessel function of the first kind, BesselK(n,x) is the modified Bessel function of the second kind. - Ilya Gutkovskiy, May 03 2016
a(n) = (hypergeom([-n,n+1],[],-1)+(-1)^n*hypergeom([-n,n+1],[],1))/2. - Peter Luschny, May 03 2016
For n >= 1, a(n) = 2^n * (theta(n, 1/2) + theta(n, -1/2))/2, where theta(n, x) = Sum_{k = 0..n} (n + k)!/((n - k)!*k!) * x^(n-k)/2^k denotes the n-th reverse Bessel polynomial. See A001497. - Peter Bala, Dec 15 2025
a(n) ~ cosh(1/2) * 2^(2*n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Feb 09 2026
EXAMPLE
a(3) = (4*3-2)*a(2)+a(1) = 10*13+2 = 132.
From Peter Bala, Dec 15 2025: (Start)
Simple continued fraction expansion of (e - 1)/(e + 1) = 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))).
The sequence of convergents begins 1/2, 1/(2 + 1/6) = 6/13, 1/(2 + 1/(6 + 1/10)) = 61/132, .... (End)
MAPLE
f:= gfun:-rectoproc({a(n)=(4*n-2)*a(n-1)+a(n-2), a(0)=1, a(1)=2}, a(n), remember):
map(f, [$0..50]); # Robert Israel, May 03 2016
MATHEMATICA
a[n_] := Sum[(2n-2k)!/((n-2k)! (2k)!), {k, 0, n/2}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 10 2019, after Vladimir Kruchinin *)
PROG
(Maxima)
a(n):=sum((2*n-2*k)!/((n-2*k)!*(2*k)!), k, 0, n/2); /* Vladimir Kruchinin, May 03 2016 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Dec 31 2002
STATUS
approved