A081920 Expansion of exp(2x)/sqrt(1-x^2).

1, 2, 5, 14, 49, 202, 1069, 6470, 48353, 391058, 3767029, 37936318, 445650385, 5359634906, 74198053661, 1036667808758, 16516851030721, 262805595346210, 4735033850606437, 84510767762583662, 1698609728377283441

Offset: 0

Author

Paul Barry, Apr 01 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A081919

Links

• Robert Israel, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A081921.

Programs

Maple

f:= gfun:-rectoproc({a(n) -2*a(n-1) -(n-1)^2*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0, a(0)=1,a(1)=2,a(2)=5},a(n),remember):
map(f, [$0..30]); # Robert Israel, Feb 19 2018

Mathematica

CoefficientList[Series[E^(2*x)/Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 04 2014 *)

Formula

E.g.f. exp(2x)/sqrt(1-x^2).

Conjecture: a(n) -2*a(n-1) -(n-1)^2*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

Conjecture confirmed using d.e. (x^2-1)*y' + (-2*x^2+x+2)*y = 0 satisfied by the E.g.f. - Robert Israel, Feb 19 2018

a(n) ~ n^n * (exp(2)+(-1)^n*exp(-2)) / exp(n). - Vaclav Kotesovec, Feb 04 2014