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A130565 Member k=6 of a family of generalized Catalan numbers. 8
1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 1166880131820, 18259838103852, 288308609783760, 4587430875645660, 73484989079268690, 1184104656043939071 (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.

For the members of the family C(k,n), k=2..9, see A130564.

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.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..806

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=6.

G.f.: inverse series of y*(1-y)^6.

a(n) = (6/7)*binomial(7*n,n)/(7*n-1). [Bruno Berselli, Jan 17 2014]

From Wolfdieter Lang, Feb 06 2020: (Start)

G.f.: (6/7)*(1 - hypergeom([-1, 1, 2, 3, 4, 5]/7, [1, 2, 3, 4, 5]/6, (7^7/6^6)*x)).

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)

MATHEMATICA

Table[Binomial[7n-2, n]/(6n-1), {n, 20}] (* Harvey P. Dale, Feb 25 2013 *)

CROSSREFS

Cf. k=5 member A130564. A006013, A006632, A118971,

Sequence in context: A246235 A213105 A138414 * A124556 A207412 A324447

Adjacent sequences:  A130562 A130563 A130564 * A130566 A130567 A130568

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jul 13 2007

STATUS

approved

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Last modified July 11 11:03 EDT 2020. Contains 335626 sequences. (Running on oeis4.)