A130565 Member k=6 of a family of generalized Catalan numbers.

1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 1166880131820, 18259838103852, 288308609783760, 4587430875645660, 73484989079268690, 1184104656043939071

Offset: 1

Author

Wolfdieter Lang, Jul 13 2007

Keywords

• nonn
• easy

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.

Crossrefs

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

Programs

Mathematica

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

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)