A189054 Expansion of e.g.f. exp(x/sqrt(1-4*x^2)).

1, 1, 1, 13, 49, 841, 6001, 126421, 1371553, 34081489, 503678881, 14391006301, 271223253841, 8751666000793, 201326507146129, 7238365225056421, 197024810845531201, 7810072695945382561, 245787442777437613633

Offset: 0

Author

Vladimir Kruchinin, Apr 16 2011

Keywords

• nonn
• changed

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A012150.

Programs

Mathematica

CoefficientList[Series[Exp[x/Sqrt[1-4*x^2]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 02 2013 *)

Maxima

a(n):= n!*sum((binomial((n-2)/2,(n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1))/k!, k,0,n);

PARI

x='x+O('x^66); /* that many terms */
egf=exp(x/sqrt(1-4*x^2)) /* = 1 +x +1/2*x^2 +13/6*x^3 +49/24*x^4 +... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 22 2011 */

Formula

a(n) = n! * Sum_{k=0..n} (binomial((n-2)/2, (n-k)/2) * 2^(n-k-1) * ((-1)^(n-k)+1))/k!.

a(n) ~ (2*n)^(n-1/3) / (sqrt(3)*exp(n-3/4*(2*n)^(1/3))). - Vaclav Kotesovec, Jun 02 2013