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A133305 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*4^i*5^(n-i), a(0) = 1.

%I A133305 #39 Sep 24 2024 02:12:00
%S A133305 1,5,45,505,6345,85405,1204245,17558705,262577745,4005148405,
%T A133305 62070886845,974612606505,15471084667545,247876665109005,
%U A133305 4003225107031845,65101209768055905,1065128963164067745,17520376884067071205,289572455530026439245,4806489064223483202905
%N A133305 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*4^i*5^(n-i), a(0) = 1.
%C A133305 Fifth column of array A103209.
%C A133305 The Hankel transform of this sequence is 20^C(n+1,2). - _Philippe Deléham_, Oct 28 2007
%H A133305 G. C. Greubel, <a href="/A133305/b133305.txt">Table of n, a(n) for n = 0..795</a>
%H A133305 Samuele Giraudo, <a href="http://arxiv.org/abs/1504.04529">Operads from posets and Koszul duality</a>, arXiv:1504.04529 [math.CO], 2015-2016.
%F A133305 G.f.: (1-z-sqrt(z^2-18*z+1))/(8*z).
%F A133305 a(n) = Sum_{k=0..n} A088617(n,k)*4^k.
%F A133305 a(n) = Sum_{k=0..n} A060693(n,k)*4^(n-k).
%F A133305 a(n) = Sum_{k=0..n} C(n+k, 2k)*4^k*C(k), C(n) given by A000108.
%F A133305 a(0) = 1, a(n) = a(n-1) + 4*Sum_{k=0..n-1} a(k)*a(n-1-k). - _Philippe Deléham_, Oct 23 2007
%F A133305 Conjecture: (n+1)*a(n) + 9*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - _R. J. Mathar_, May 23 2014
%F A133305 G.f.: 1/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, May 10 2017
%F A133305 a(n) = hypergeom([-n, n + 1], [2], -4). - _Peter Luschny_, Jan 08 2018
%F A133305 a(n) ~ 5^(1/4) * phi^(6*n + 3) / (2^(5/2) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Nov 21 2021
%t A133305 a[n_] := Hypergeometric2F1[-n, n + 1, 2, -4];
%t A133305 Table[a[n], {n, 0, 16}] (* _Peter Luschny_, Jan 08 2018 *)
%t A133305 CoefficientList[Series[(1-x-Sqrt[x^2-18*x+1])/(8*x), {x, 0, 50}], x] (* _G. C. Greubel_, Feb 10 2018 *)
%o A133305 (PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-18*x+1))/(8*x)) \\ _G. C. Greubel_, Feb 10 2018
%o A133305 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-18*x+1))/(8*x))) // _G. C. Greubel_, Feb 10 2018
%Y A133305 Cf. A000108, A060693, A103209, A103210, A103211.
%K A133305 nonn
%O A133305 0,2
%A A133305 _Philippe Deléham_, Oct 18 2007