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a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3.
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%I #24 Aug 19 2019 17:23:32

%S 1,3,6,21,78,318,1356,5997,27222,126138,594132,2836290,13692300,

%T 66729180,327855768,1622216829,8076311142,40427919714,203353800324,

%U 1027318915254,5210182030308,26517609163812,135397544040744,693364054299474

%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3.

%H Vincenzo Librandi, <a href="/A025229/b025229.txt">Table of n, a(n) for n = 1..200</a>

%F G.f.: (1-sqrt(1-4*x-8*x^2))/2. - _Michael Somos_, Jun 08 2000

%F a(n) = Sum_{k=0..n} 2^(n-k)*C(k)*C(k+1, n-k) [offset 0]. - _Paul Barry_, Feb 22 2005

%F Another recurrence formula: n*a(n) = (4*n-6)*a(n-1)+(8*n-24)*a(n-2). - _Richard Choulet_, Dec 16 2009

%F a(n) ~ sqrt(3-sqrt(3))*(2+2*sqrt(3))^n/(4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012

%p A025229 := proc(n)

%p option remember;

%p if n <=1 then

%p n;

%p elif n = 2 then

%p 3;

%p else

%p add( procname(n-i)*procname(i),i=1..n-1) ;

%p end if;

%p end proc: # _R. J. Mathar_, Jun 17 2015

%t Table[SeriesCoefficient[(1-Sqrt[1-4*x-8*x^2])/2,{x,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)

%o (PARI) a(n)=polcoeff((1-sqrt(1-4*x-8*x^2+x*O(x^n)))/2,n)

%K nonn,easy

%O 1,2

%A _Clark Kimberling_