A143669 a(n) = binomial((n+1)^2, n) / (n+1)^2.

1, 1, 4, 35, 506, 10472, 285384, 9706503, 397089550, 19022318084, 1045659267016, 64924369564353, 4496010926381352, 343688726144945040, 28753733905585301136, 2613784129155164386575, 256569498889138342791510, 27050758656206146528056236

Offset

0,3

Comments

From Peter Bala, Dec 02 2015: (Start)

Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n/(n^2 + 1)). Cf. A135862.

See Armstrong et al. for combinatorial interpretations of these generalized Catalan number sequences. (End)

Links

Seiichi Manyama, Table of n, a(n) for n = 0..339

D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Mathematica

Table[Binomial[(n + 1)^2, n]/(n + 1)^2, {n, 0, 30}] (* Vincenzo Librandi, Dec 09 2015 *)

Prog

(PARI) a(n)=binomial((n+1)^2, n)/(n+1)^2
(Magma) [Binomial((n+1)^2, n) / (n+1)^2: n in [0..20]]; // Vincenzo Librandi, Dec 09 2015

Crossrefs

Cf. A295765, A135862, A182316.

Sequence in context: A287886 A342207 A224797 * A349656 A180716 A171778

Adjacent sequences: A143666 A143667 A143668 * A143670 A143671 A143672

Keyword

nonn,easy

Author

Paul D. Hanna, Aug 28 2008

Status

approved