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A025229 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 1,3.

%I A025229 #27 Feb 07 2025 00:34:15
%S A025229 1,3,6,21,78,318,1356,5997,27222,126138,594132,2836290,13692300,
%T A025229 66729180,327855768,1622216829,8076311142,40427919714,203353800324,
%U A025229 1027318915254,5210182030308,26517609163812,135397544040744,693364054299474
%N A025229 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 1,3.
%H A025229 Vincenzo Librandi, <a href="/A025229/b025229.txt">Table of n, a(n) for n = 1..200</a>
%F A025229 G.f.: (1-sqrt(1-4*x-8*x^2))/2. - _Michael Somos_, Jun 08 2000
%F A025229 a(n) = Sum_{k=0..n} 2^(n-k)*C(k)*C(k+1, n-k) [offset 0]. - _Paul Barry_, Feb 22 2005
%F A025229 Another recurrence formula: n*a(n) = (4*n-6)*a(n-1)+(8*n-24)*a(n-2). - _Richard Choulet_, Dec 16 2009
%F A025229 a(n) ~ sqrt(3-sqrt(3))*(2+2*sqrt(3))^n/(4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012
%p A025229 A025229 := proc(n)
%p A025229     option remember;
%p A025229     if n <=1 then
%p A025229         n;
%p A025229     elif n = 2 then
%p A025229         3;
%p A025229     else
%p A025229         add( procname(n-i)*procname(i),i=1..n-1) ;
%p A025229     end if;
%p A025229 end proc: # _R. J. Mathar_, Jun 17 2015
%t A025229 Table[SeriesCoefficient[(1-Sqrt[1-4*x-8*x^2])/2,{x,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)
%o A025229 (PARI) a(n)=polcoeff((1-sqrt(1-4*x-8*x^2+x*O(x^n)))/2,n)
%K A025229 nonn,easy
%O A025229 1,2
%A A025229 _Clark Kimberling_
%E A025229 Name clarified by _Robert C. Lyons_, Feb 06 2025