A081921 Expansion of exp(3x)/sqrt(1-x^2).

1, 3, 10, 36, 144, 648, 3384, 20520, 145728, 1181952, 10917504, 111601152, 1265777280, 15544566144, 208320719616, 2980582728192, 46020833427456, 751100760576000, 13121167636058112, 240473024248393728, 4687531209011183616, 95293672221284794368

Offset

0,2

Comments

Binomial transform of A081920.

Links

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

Formula

E.g.f. exp(3x)/sqrt(1-x^2)

a(n) = 3^n*n!*Sum_{k=0..floor(n/2)} binomial(2*k, k)/(n-2*k)!/36^k. - Vladeta Jovovic, Oct 11 2003

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

a(n) ~ (exp(6)+(-1)^n)*n^n/exp(n+3). - Vaclav Kotesovec, Oct 05 2013

Mathar's conjecture follows from the differential equation (-3*x^2+x+3)*y+(x^2-1)*y'=0 satisfied by the E.g.f. - Robert Israel, Mar 14 2019

Maple

f:= gfun:-rectoproc({a(n)-3*a(n-1)-(n-1)^2*a(n-2)+(3*(n-1))*(n-2)*a(n-3), a(0)=1, a(1)=3, a(2)=10}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 14 2019

Mathematica

CoefficientList[Series[E^(3*x)/Sqrt[1-x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)

Crossrefs

Cf. A081922.

Sequence in context: A129247 A162162 A149042 * A353262 A165792 A010373

Adjacent sequences: A081918 A081919 A081920 * A081922 A081923 A081924

Keyword

easy,nonn

Author

Paul Barry, Apr 01 2003

Status

approved