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A133307 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*6^i*7^(n-i), a(0)=1.

%I A133307 #28 Sep 08 2022 08:45:31
%S A133307 1,7,91,1477,26845,522739,10663471,224939113,4866571801,107393779423,
%T A133307 2407939176643,54700070934061,1256249370578293,29119953189833611,
%U A133307 680401905145643863,16008309928027493713,378930780842531820721,9017843351806985482423,215634517504141993966891
%N A133307 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*6^i*7^(n-i), a(0)=1.
%C A133307 Seventh column of array A103209.
%C A133307 The Hankel transform of this sequence is 42^C(n+1,2). - _Philippe Deléham_, Oct 28 2007
%H A133307 G. C. Greubel, <a href="/A133307/b133307.txt">Table of n, a(n) for n = 0..700</a>
%F A133307 G.f.: (1-z-sqrt(z^2-26*z+1))/(12*z).
%F A133307 a(n) = Sum_{k=0..n} A088617(n,k)*6^k .
%F A133307 a(n) = Sum_{k=0..n} A060693(n,k)*6^(n-k).
%F A133307 a(n) = Sum_{k=0..n} C(n+k, 2k)6^k*C(k), C(n) given by A000108.
%F A133307 a(0)=1, a(n) = a(n-1) + 6*Sum_{k=0..n-1} a(k)*a(n-1-k). - _Philippe Deléham_, Oct 23 2007
%F A133307 Conjecture: (n+1)*a(n) + 13*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - _R. J. Mathar_, May 23 2014
%F A133307 a(n) = hypergeom([-n, n+1], [2], -6). # _Peter Luschny_, May 23 2014
%F A133307 G.f.: 1/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, May 10 2017
%F A133307 a(n) ~ 42^(1/4) * (13 + 2*sqrt(42))^(n + 1/2) / (12*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Nov 29 2021
%p A133307 a := n -> hypergeom([-n, n+1], [2], -6);
%p A133307 seq(round(evalf(a(n),32)),n=0..16); # _Peter Luschny_, May 23 2014
%t A133307 CoefficientList[Series[(1-x-Sqrt[x^2-26*x+1])/(12*x), {x,0,50}], x] (* _G. C. Greubel_, Feb 10 2018 *)
%o A133307 (PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-26*x+1))/(12*x)) \\ _G. C. Greubel_, Feb 10 2018
%o A133307 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-26*x+1))/(12*x))) // _G. C. Greubel_, Feb 10 2018
%Y A133307 Cf. A000108, A060693, A103209, A103210, A103211.
%K A133307 nonn
%O A133307 0,2
%A A133307 _Philippe Deléham_, Oct 18 2007