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A133306 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*5^i*6^(n-i), a(0)=1.

%I A133306 #24 Sep 08 2022 08:45:31
%S A133306 1,6,66,906,13926,229326,3956106,70572066,1291183806,24095736726,
%T A133306 456879955026,8776867331706,170459895028566,3341423256586206,
%U A133306 66023812564384026,1313634856606430226,26295597219228901806,529199848207277494566,10701116421278640683106,217317899302044152030826
%N A133306 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*5^i*6^(n-i), a(0)=1.
%C A133306 Sixth column of array A103209.
%C A133306 The Hankel transform of this sequence is 30^C(n+1,2). - _Philippe Deléham_, Oct 28 2007
%H A133306 G. C. Greubel, <a href="/A133306/b133306.txt">Table of n, a(n) for n = 0..745</a>
%F A133306 G.f.: (1-z-sqrt(z^2-22*z+1))/(10*z).
%F A133306 a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*5^k.
%F A133306 a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*5^(n-k).
%F A133306 a(n) = Sum_{k, 0<=k<=n} C(n+k, 2*k) 5^k*C(k), C(n) given by A000108.
%F A133306 a(0)=1, a(n) = a(n-1) + 5*Sum_{k=0..n-1} a(k)*a(n-1-k). - _Philippe Deléham_, Oct 23 2007
%F A133306 Conjecture: (n+1)*a(n) + 11*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - _R. J. Mathar_, May 23 2014
%F A133306 G.f.: 1/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, May 10 2017
%F A133306 a(n) ~ 3^(1/4) * (11 + 2*sqrt(30))^(n + 1/2) / (10^(3/4) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 29 2021
%t A133306 CoefficientList[Series[(1-x-Sqrt[x^2-22*x+1])/(10*x), {x,0,50}], x] (* _G. C. Greubel_, Feb 10 2018 *)
%o A133306 (PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-22*x+1))/(10*x)) \\ _G. C. Greubel_, Feb 10 2018
%o A133306 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-22*x+1))/(10*x))) // _G. C. Greubel_, Feb 10 2018
%Y A133306 Cf. A000108, A060693, A103209, A103210, A103211.
%K A133306 nonn
%O A133306 0,2
%A A133306 _Philippe Deléham_, Oct 18 2007