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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.

Elżbieta Liszewska, Wojciech Młotkowski, 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.)