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, 19180066927056942918, 312128008052605459000, 5100701804712883789200

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 and 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)

a(n) ~ 7^(7*n-3/2) / (3^(6*n-1/2) * 64^n * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 14 2025

D-finite with recurrence 72*n*(6*n-5)*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n) -7*(7*n-3)*(7*n-6)*(7*n-2)*(7*n-5)*(7*n-8)*(7*n-4)*a(n-1)=0. - R. J. Mathar, May 18 2026

Mathematica

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

Crossrefs

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

Sequence in context: A213105 A138414 A349363 * A124556 A369514 A379023

Adjacent sequences: A130562 A130563 A130564 * A130566 A130567 A130568

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 13 2007

Status

approved