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A025230 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,1.

%I A025230 #46 Aug 17 2025 07:48:25
%S A025230 3,1,6,37,234,1514,9996,67181,458562,3172478,22206420,157027938,
%T A025230 1120292388,8055001716,58314533400,424740506109,3110401363122,
%U A025230 22888001498102,169155516667524,1255072594261142,9345400450314924,69812926066668044,523072984217339304
%N A025230 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 3,1.
%H A025230 Vincenzo Librandi, <a href="/A025230/b025230.txt">Table of n, a(n) for n = 1..200</a>
%H A025230 Rigoberto Flórez, Leandro Junes, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez/florez4.html">Further Results on Paths in an n-Dimensional Cubic Lattice</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
%F A025230 G.f.: (1-sqrt(1-12*x+32*x^2))/2. - _Michael Somos_, Jun 08 2000
%F A025230 D-finite with recurrence n*a(n) = (12*n-18)*a(n-1) - 32*(n-3)*a(n-2) - _Richard Choulet_, Dec 17 2009
%F A025230 a(n) ~ 2^(3*n-5/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 11 2013
%F A025230 a(n) = 4^(n-2)*hypergeom([3/2, -n+2], [3], -1) for n>1. - _Peter Luschny_, Feb 03 2015
%F A025230 a(n+1) = GegenbauerC(n-1, -n, -3)/n for n>=1. - _Peter Luschny_, May 09 2016
%F A025230 From _Peter Bala_, Feb 03 2024: (Start)
%F A025230 G.f.: 3*x + x^2/(1 - 4*x) * c(x/(1 - 4*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
%F A025230 a(n+2) = Sum_{k = 0..n} 4^(n-k)*binomial(n, k)*Catalan(k+1).
%F A025230 G.f.: 3*x + x^2/(1 - 8*x) * c(-x/(1 - 8*x))^2.
%F A025230 a(n+2) = 8^n * Sum_{k = 0..n} (-8)^(-k)*binomial(n, k)*Catalan(k+1).
%F A025230 a(n+2) = 8^n * hypergeom([-n, 3/2], [3], 1/2).
%F A025230 a(n) is odd iff n is a power of 2. (End)
%p A025230 h := n -> simplify(4^n*hypergeom([3/2, -n], [3], -1)):
%p A025230 a := n -> `if`(n=1, 3, h(n-2)):
%p A025230 seq(a(n), n=1..21); # _Peter Luschny_, Feb 03 2015
%t A025230 Rest[CoefficientList[Series[(1-Sqrt[1-12x+32x^2])/2,{x,0,30}],x]]  (* _Harvey P. Dale_, Feb 22 2011 *)
%o A025230 (PARI) a(n)=polcoeff((1-sqrt(1-12*x+32*x^2+x*O(x^n)))/2,n)
%o A025230 (PARI) {a(n)=if(n<2, 3*(n==1), n--; polcoeff( serreverse( x/(1+6*x+x^2) +x*O(x^n) ), n))} /* _Michael Somos_, Oct 14 2006 */
%Y A025230 For Sum_{k = 0..n} m^(n-k)*binomial(n, k)*Catalan(k+1) see A126120 (m = -2), A001006 (m = -1), A000108 (m = 0), A002212 (m = 1), A005572 (m = 2), A182401 (m = 3), A025230 (m = 4).
%K A025230 nonn,easy
%O A025230 1,1
%A A025230 _Clark Kimberling_
%E A025230 Name clarified by _Robert C. Lyons_, Feb 06 2025