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%I #16 Sep 08 2022 08:44:49
%S 1,1,1,2,6,17,50,150,458,1420,4460,14165,45422,146870,478366,1568076,
%T 5169332,17127466,57005220,190503772,638986740,2150472248,7259433880,
%U 24574658829,83404648534,283742556262,967415888822,3305124188428
%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5.
%H Michael De Vlieger, <a href="/A025272/b025272.txt">Table of n, a(n) for n = 1..1000</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%F Conjecture: n*a(n) +(n+1)*a(n-1) +10*(-2*n+5)*a(n-2) +2*(2*n-9)*a(n-3) +2*(12*n-67)*a(n-4) +20*(n-7)*a(n-5)=0. - _R. J. Mathar_, Nov 21 2014
%F G.f.: 1/2 - sqrt(4*x^4+4*x^3-4*x+1)/2. - _Vaclav Kotesovec_, Jan 25 2015
%F Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 2*(2*n-9)*a(n-3) - 4*(n-6)*a(n-4). - _Vaclav Kotesovec_, Jan 25 2015
%t nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 1; aa[[3]] = 1; aa[[4]] = 2; Do[aa[[n]] = Sum[aa[[k]]*aa[[n-k]],{k,1,n-1}],{n,5,nmax}]; aa (* _Vaclav Kotesovec_, Jan 25 2015 *)
%t Rest@ CoefficientList[Series[1/2 - Sqrt[4 x^4 + 4 x^3 - 4 x + 1]/2, {x, 0, 28}], x] (* _Michael De Vlieger_, Jan 02 2020 *)
%o (Magma) a:=[1,1,1,2]; for n in [5..30] do Append(~a,&+[a[k]*a[n-k]:k in [1..n-1]] ); end for; a; // _Marius A. Burtea_, Jan 02 2020
%K nonn
%O 1,4
%A _Clark Kimberling_
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