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%I #25 Mar 07 2024 04:23:34
%S 1,5,40,385,4095,46376,548340,6690585,83615350,1064887395,13770292256,
%T 180320238280,2386316821325,31864803599700,428798445360120,
%U 5809228810425801,79168272296871450,1084567603590147950
%N Member k=5 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 The family c(k,n):=binomial((k+1)*n-2,n)/(k*n-1), n>=1, has the members A000108, A006013, A006632, A118971 for k=1,2,3,4, respectively (but the offset there is 0).
%C The members of the C(k,n) family for positive k are: A000012 (powers of 1), A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994, for k=1..9.
%C The ordinary generating functions for the k-family {c(k, n+1)}_{n>=0}, k >= 1, are G(k, x) = hypergeometric(Aseq(k+1), Bseq(k), ((k+1)^(k+1)/k^k)*x), with Aseq(k+1) = [a(k)_1,..., a(k)_{k+1}], where a(k)_j = (2*k - (j-1))/(k+1), and Bseq(k) = [b(k)_1, ..., b(k)_k], where b(k)_j = (2*k - (j-1))/k. The e.g.f. has an extra 1 in the B-section, which leads to a cancellation with the A-section 1 term. Thanks to Dixon J. Jones for asking for the general formulas. - _Wolfdieter Lang_, Feb 04 2024
%C The o.g.f. of {C(k, n)}_{n>=0} is in the Graham-Knuth-Patashnik book denoted as B_k(z) on pp. 200, 349 (2nd ed. 1994, pp. 200, 363). - _Wolfdieter Lang_, Mar 07 2024
%D Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1994, pp. 200, 363.
%H Michael De Vlieger, <a href="/A130564/b130564.txt">Table of n, a(n) for n = 1..856</a>
%H K. Kobayashi, H. Morita and M. Hoshi, <a href="https://doi.org/10.1109/ISIT.2000.866305">Coding of ordered trees</a>, Proceedings, IEEE International Symposium on Information Theory, ISIT 2000, Sorrento, Italy, Jun 25 2000.
%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=5.
%F G.f.: inverse series of y*(1-y)^5.
%F a(n) = (5/6)*binomial(6*n,n)/(6*n-1). [_Bruno Berselli_, Jan 17 2014]
%F From _Wolfdieter Lang_, Feb 06 2020: (Start)
%F G.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4]/5,(6^6/5^5)*x)).
%F E.g.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4, 5]/5,(6^6/5^5)*x)). (End)
%F D-finite with recurrence 5*n*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-7)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n-5)*a(n-1)=0. - _R. J. Mathar_, May 07 2021
%t Rest@ CoefficientList[InverseSeries[Series[y (1 - y)^5, {y, 0, 18}], x], x] (* _Michael De Vlieger_, Oct 13 2019 *)
%Y Cf. A000012, A000108, A001764, A002293, A002294, A002295, A002296, A006013, A062994, A006632, A007556, A118971, A130565, A234466, A234513, A234573, A235340.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Jul 13 2007
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