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A014432 a(n) = Sum_{i=1..n-1} a(i)*a(n-1-i), with a(0) = 1, a(1) = 3.

%I A014432 #42 Dec 12 2022 20:52:58
%S A014432 1,3,3,12,30,111,363,1353,4917,18777,71769,280506,1103556,4395009,
%T A014432 17622309,71220828,289510662,1183627137,4862148753,20061888924,
%U A014432 83100910530,345457823493,1440734205513,6026408186457,25275954499905,106277040064191
%N A014432 a(n) = Sum_{i=1..n-1} a(i)*a(n-1-i), with a(0) = 1, a(1) = 3.
%H A014432 Robert Israel, <a href="/A014432/b014432.txt">Table of n, a(n) for n = 0..1545</a>
%F A014432 G.f.: ((1+x-sqrt(1-2*x-11*x^2)))/(2*x). - _Michael Somos_, Jun 08 2000; corrected by _Robert Israel_, Sep 10 2020
%F A014432 a(n) = (3/(11*n)) * ((3+n)*A025237(n+1) - (2*n+3)*A025237(n)) for n>0. [_Mark van Hoeij_, Jul 02 2010]
%F A014432 (n+1)*a(n) = (2*n-1)*a(n-1)+11*(n-2)*a(n-2). - _Robert Israel_, Sep 10 2020
%F A014432 G.f.: 1 + 3*x/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - x - 3*x^2/(1 - ...))))) (continued fraction). - _Nikolaos Pantelidis_, Nov 24 2022
%p A014432 seq(coeff(convert(series((1+x-sqrt(1-2*x-11*x^2))/(2*x),x,50),polynom),x,i),i=0..30);
%p A014432 A014431:=proc(n) options remember: local i: if n<2 then RETURN([1,3][n+1]) else RETURN(add(A014431(i)*A014431(n-1-i),i=1..n-1)) fi:end;seq(A014431(n),n=0..30); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004
%t A014432 Rest[CoefficientList[Series[(1+x-Sqrt[1-2x-11x^2])/2,{x,0,30}],x]] (* _Harvey P. Dale_, Apr 17 2019 *)
%o A014432 (PARI) a(n)=polcoeff((1+x-sqrt(1-2*x-11*x^2+x*O(x^n)))/2,n)
%Y A014432 Cf. A025237.
%K A014432 nonn
%O A014432 0,2
%A A014432 _Wouter Meeussen_
%E A014432 Corrected by C. Ronaldo (aga_new_ac(AT)hotmail.com) and _Ralf Stephan_, Dec 19 2004