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A133308 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*7^i*8^(n-i), a(0)=1.

%I A133308 #24 Sep 08 2022 08:45:31
%S A133308 1,8,120,2248,47160,1059976,24958200,607693640,15175702200,
%T A133308 386555020552,10004252294520,262321706465736,6953918939056440,
%U A133308 186059575955360136,5018045415643478520,136276936332343342152,3723442515218861494200,102281105054908404972040
%N A133308 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*7^i*8^(n-i), a(0)=1.
%C A133308 Eighth column of array A103209.
%C A133308 The Hankel transform of this sequence is 56^C(n+1,2). - _Philippe Deléham_, Oct 28 2007
%H A133308 G. C. Greubel, <a href="/A133308/b133308.txt">Table of n, a(n) for n = 0..675</a>
%F A133308 G.f.: (1-z-sqrt(z^2-30*z+1))/(14*z).
%F A133308 a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*7^k.
%F A133308 a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*7^(n-k).
%F A133308 a(n) = Sum_{k, 0<=k<=n} C(n+k, 2k)7^k*C(k), C(n) given by A000108.
%F A133308 a(0)=1, a(n) = a(n-1) + 7*Sum_{k=0..n-1} a(k)*a(n-1-k). - _Philippe Deléham_, Oct 23 2007
%F A133308 Conjecture: (n+1)*a(n) + 15*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - _R. J. Mathar_, May 23 2014
%F A133308 a(n) = hypergeom([-n, n+1], [2], -7). - _Peter Luschny_, May 23 2014
%F A133308 G.f.: 1/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, May 10 2017
%p A133308 a := n -> hypergeom([-n, n+1], [2], -7);
%p A133308 seq(round(evalf(a(n), 32)), n=0..15); # _Peter Luschny_, May 23 2014
%t A133308 CoefficientList[Series[(1-x-Sqrt[x^2-30*x+1])/(14*x), {x,0,50}], x] (* _G. C. Greubel_, Feb 10 2018 *)
%o A133308 (PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-30*x+1))/(14*x)) \\ _G. C. Greubel_, Feb 10 2018
%o A133308 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-30*x+1))/(14*x))) // _G. C. Greubel_, Feb 10 2018
%Y A133308 Cf. A000108, A060693, A103209, A103210, A103211.
%K A133308 nonn
%O A133308 0,2
%A A133308 _Philippe Deléham_, Oct 18 2007