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%I #14 Feb 06 2020 19:21:06
%S 1,6,57,650,8184,109668,1533939,22137570,327203085,4928006512,
%T 75357373305,1166880131820,18259838103852,288308609783760,
%U 4587430875645660,73484989079268690,1184104656043939071
%N Member k=6 of a family of generalized Catalan numbers.
%C The generalized Catalan numbers C(k,n):= binomial(k*n+1,n)/(k*n+1) become for negative k=-|k|, with |k|>=2, ((-1)^(n-1))*binomial((|k|+1)*n-2,n)/(|k|*n-1), n>=0.
%C For the members of the family C(k,n), k=2..9, see A130564.
%C The family c(k,n):=binomial((k+1)*n-2,n)/(k*n-1), n>=1, has the members A006013, A006632, A118971,for k=2,3,4 respectively (but the offset there is 0) and A130564 for k=5.
%H Harvey P. Dale, <a href="/A130565/b130565.txt">Table of n, a(n) for n = 1..806</a>
%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F a(n) = binomial((k+1)*n-2,n)/(k*n-1), with k=6.
%F G.f.: inverse series of y*(1-y)^6.
%F a(n) = (6/7)*binomial(7*n,n)/(7*n-1). [_Bruno Berselli_, Jan 17 2014]
%F From _Wolfdieter Lang_, Feb 06 2020: (Start)
%F G.f.: (6/7)*(1 - hypergeom([-1, 1, 2, 3, 4, 5]/7, [1, 2, 3, 4, 5]/6, (7^7/6^6)*x)).
%F E.g.f.: (6/7)*(1 - hypergeom([-1, 1, 2, 3, 4, 5]/7, [1, 2, 3, 4, 5, 6]/6, (7^7/6^6)*x)). (End)
%t Table[Binomial[7n-2,n]/(6n-1),{n,20}] (* _Harvey P. Dale_, Feb 25 2013 *)
%Y Cf. k=5 member A130564. A006013, A006632, A118971,
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Jul 13 2007
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