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a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*7^i*8^(n-i), a(0)=1.
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%I #24 Sep 08 2022 08:45:31

%S 1,8,120,2248,47160,1059976,24958200,607693640,15175702200,

%T 386555020552,10004252294520,262321706465736,6953918939056440,

%U 186059575955360136,5018045415643478520,136276936332343342152,3723442515218861494200,102281105054908404972040

%N 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 Eighth column of array A103209.

%C The Hankel transform of this sequence is 56^C(n+1,2). - _Philippe Deléham_, Oct 28 2007

%H G. C. Greubel, <a href="/A133308/b133308.txt">Table of n, a(n) for n = 0..675</a>

%F G.f.: (1-z-sqrt(z^2-30*z+1))/(14*z).

%F a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*7^k.

%F a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*7^(n-k).

%F a(n) = Sum_{k, 0<=k<=n} C(n+k, 2k)7^k*C(k), C(n) given by A000108.

%F 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 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 a(n) = hypergeom([-n, n+1], [2], -7). - _Peter Luschny_, May 23 2014

%F 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 a := n -> hypergeom([-n, n+1], [2], -7);

%p seq(round(evalf(a(n), 32)), n=0..15); # _Peter Luschny_, May 23 2014

%t CoefficientList[Series[(1-x-Sqrt[x^2-30*x+1])/(14*x), {x,0,50}], x] (* _G. C. Greubel_, Feb 10 2018 *)

%o (PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-30*x+1))/(14*x)) \\ _G. C. Greubel_, Feb 10 2018

%o (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 Cf. A000108, A060693, A103209, A103210, A103211.

%K nonn

%O 0,2

%A _Philippe Deléham_, Oct 18 2007