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%I #24 Jan 09 2023 10:39:23
%S 1,4,4,20,52,228,804,3444,13780,59588,253252,1113556,4892276,21860260,
%T 98055780,444148020,2021194260,9257373060,42583930500,196811777940,
%U 913015265460,4251135572580,19856669967780,93027410579700,436999575464532,2057978301836868,9713953354107844
%N a(n) = Sum_{i=0..n-1} a(i)*a(n-i), with a(0) = 1 and a(1) = 4.
%H Vincenzo Librandi, <a href="/A014433/b014433.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1+x-sqrt(1-2*x-15*x^2))/(2*x) - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004
%F a(n) = (-3)^n*(hypergeom([1/2, n+1],[1],8/5)-5*hypergeom([1/2, n],[1],8/5))*(-15)^(1/2)/(10*(n+1)) for n>0. - _Mark van Hoeij_, Jul 02 2010
%F Recurrence: (n+1)*a(n) = (2*n-1)*a(n-1) + 15*(n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 07 2012
%F a(n) ~ 5^(n+1/2)/(sqrt(2*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012
%F a(n) = (-2)^(n+1)*C(2*n,n-1)*hypergeom([-n-1,-n+1],[-n+1/2],5/8)/n for n>=1. - _Peter Luschny_, May 08 2016
%F a(n) = 2^(n+1)*GegenbauerC(n-1,-n,-1/4)/n for n>=1. - _Peter Luschny_, May 08 2016
%F G.f.: 1 + 4*x/G(x) with G(x) = (1 - x - 4*x^2/G(x)) (continued fraction). - _Nikolaos Pantelidis_, Jan 09 2023
%p a := n -> `if`(n=0,1,simplify(2^(n+1)*GegenbauerC(n-1,-n,-1/4)/n)):
%p seq(a(n), n=0..26); # _Peter Luschny_, May 08 2016
%t Table[SeriesCoefficient[(1+x-Sqrt[1-2*x-15*x^2])/(2*x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)
%t nxt[{n_,a_,b_}]:={n+1,b,(b(2n+1)+15a(n-1))/(n+2)}; NestList[nxt,{1,1,4},30][[All,2]] (* _Harvey P. Dale_, Jul 07 2019 *)
%o (PARI) x='x+O('x^66); Vec((1+x-sqrt(1-2*x-15*x^2))/(2*x)) \\ _Joerg Arndt_, May 04 2013
%K nonn
%O 0,2
%A _Wouter Meeussen_
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