A182316 -id:A182316 - cp's OEIS frontend

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

Paul D. Hanna, Aug 28 2008

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)

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.

Cf. A295765, A135862, A182316.

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

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

PARI
```
a(n)=binomial((n+1)^2,n)/(n+1)^2
```

A351501 a(n) = binomial(n^2 + n - 1, n) / (n^2 + n - 1).

Original entry on oeis.org

1, 2, 15, 204, 4095, 109668, 3689595, 149846840, 7141879503, 391139588190, 24218296445200, 1673538279265020, 127715832778905150, 10670643284149377480, 968929726650218004435, 95024894699780159868144, 10011211830149283223044015

Offset: 1

F. Chapoton, May 03 2022

nonn

Empirical: In the ring of symmetric functions over the fraction field Q(q, t), let s(n) denote the Schur function indexed by n. Then (up to sign) a(n) is the coefficient of s(1^n) in nabla^(n) s(n) with q=t=1, where nabla denotes the "nabla operator" on symmetric functions.

Closely related to A177784. See also A091144.

Diagonal of A162382. Multiple of A182316.

Table[With[{c=n^2+n-1},Binomial[c,n]/c],{n,20}] (* Harvey P. Dale, Jan 01 2024 *)

from math import comb
def A351501(n): return comb(m := n**2+n-1,n)//m # Chai Wah Wu, May 07 2022

[binomial(n*n+n-1,n)/(n*n+n-1) for n in range(1,29)]

a(n) ~ c*exp(n-1/(6*n))*n^(n-5/2), where c = sqrt(e/(2*Pi)). - Stefano Spezia, May 04 2022

a(n) = n * A182316(n - 1). - F. Chapoton, Sep 22 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI