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Expansion of e.g.f. 1/sqrt(1-2x-2x^2).
1

%I #24 May 10 2020 19:33:50

%S 1,1,5,33,321,3945,59445,1056825,21677985,503799345,13084021125,

%T 375524312625,11803392302625,403235809601625,14876913457531125,

%U 589498927632239625,24969077812488434625,1125803018759825030625

%N Expansion of e.g.f. 1/sqrt(1-2x-2x^2).

%F a(n) = (n!/2^n)*A084609(n);

%F a(n) = (n!/2^n) * Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(2(n-k),n)*2^k;

%F a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(2(n-k),n)*2^(k-n).

%F D-finite with recurrence: a(n) +(1-2*n)*a(n-1) -2*(n-1)^2*a(n-2)=0. - _R. J. Mathar_, Nov 15 2011

%F a(n) ~ 2^(n+1/2)*n^n/(sqrt(3-sqrt(3))*exp(n)*(sqrt(3)-1)^n). - _Vaclav Kotesovec_, Jun 26 2013

%t CoefficientList[Series[1/Sqrt[1-2*x-2*x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 26 2013 *)

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(1/sqrt(1-2*x-2*x^2))) \\ _Michel Marcus_, May 10 2020

%Y Cf. A012244.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Sep 08 2004