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 A130564 Member k=5 of a family of generalized Catalan numbers. 11
 1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 180320238280, 2386316821325, 31864803599700, 428798445360120, 5809228810425801, 79168272296871450, 1084567603590147950 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. 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). 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=2..9. LINKS Michael De Vlieger, Table of n, a(n) for n = 1..856 K. Kobayashi, H. Morita and M. Hoshi, Coding of ordered trees, Proceedings, IEEE International Symposium on Information Theory, ISIT 2000, Sorrento, Italy, Jun 25 2000. ElÅ¼bieta Liszewska, Wojciech MÅ‚otkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019. FORMULA a(n) = binomial((k+1)*n-2,n)/(k*n-1), with k=5. G.f.: inverse series of y*(1-y)^5. a(n) = (5/6)*binomial(6*n,n)/(6*n-1). [Bruno Berselli, Jan 17 2014] From Wolfdieter Lang, Feb 06 2020: (Start) G.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4]/5,(6^6/5^5)*x)). 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) MATHEMATICA Rest@ CoefficientList[InverseSeries[Series[y (1 - y)^5, {y, 0, 18}], x], x] (* Michael De Vlieger, Oct 13 2019 *) CROSSREFS Cf. A000012, A000108, A001764, A002293, A002294, A002295, A002296, A006013, A062994, A006632, A007556, A118971. Sequence in context: A219560 A271957 A220673 * A124555 A152601 A079158 Adjacent sequences:  A130561 A130562 A130563 * A130565 A130566 A130567 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Jul 13 2007 STATUS approved

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Last modified August 3 11:40 EDT 2020. Contains 336198 sequences. (Running on oeis4.)