A028329 Twice central binomial coefficients.

2, 4, 12, 40, 140, 504, 1848, 6864, 25740, 97240, 369512, 1410864, 5408312, 20801200, 80233200, 310235040, 1202160780, 4667212440, 18150270600, 70690527600, 275693057640, 1076515748880, 4208197927440, 16466861455200, 64495207366200, 252821212875504, 991837065896208

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Comments

Central elements in the even-Pascal triangle A028326.

If Y is a 3-subset of an 2n-set X then, for n>=3, a(n-1) is the number of (n+1)-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007

a(n) denotes the number of ways one can reach the (n,n) point in an n X n grid via the point (n-1, n-1) starting from (0,0) when moving right and up is allowed [From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 29 2009]

It appears that a(n-1) is also the number of quivers in the mutation class of twisted types BD_n and CD_n for n >= 3. - Christian Stump, Nov 03 2010

This is the case m = n+1 in the Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - Bruno Berselli, Apr 27 2012

From Ran Pan, Feb 01 2016: (Start)

a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that bounce off the diagonal y = x an even number of times. Details can be found in Section 4.2 in Pan and Remmel's link.

a(n) is the number of North-East paths from (0,0) to (n+1,n+1) that cross the diagonal y = x an even number of times. Details can be found in Section 4.3 in Pan and Remmel's link. (End)

References

• Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third Edition), page 11.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Crossrefs

Bisection of A047073, A063886.

First differences of A054113.

Cf. A000984, A095660, A100320, A162551.

Cf. A028326, A028327, A028328, A028330, A028331, A028332.

Programs

Magma

[2*(n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Jul 13 2024

Maple

seq(add(binomial(2*n,n),k=1..2),n=0..23); # Zerinvary Lajos, Dec 14 2007

Mathematica

Table[2Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, Aug 08 2011 *)

PARI

a(n)=2*binomial(2*n,n)

SageMath

[2*binomial(2*n,n) for n in range(31)] # G. C. Greubel, Jul 13 2024

Formula

G.f.: 2/sqrt(1 - 4*x).

a(n) = 2*A000984(n).

a(n) = 2 * binomial(2*n, n).

a(n) = A100320(n) = A095660(2*n,n) for n > 0. - Reinhard Zumkeller, Apr 08 2012

G.f.: G(0), where G(k)= 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) + (k + 1)/ G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013

a(n) = binomial(2*n+2, n+1) - A162551(n). - Ran Pan, Feb 01 2016

D-finite with recurrence: n*a(n) + 2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

E.g.f.: 2*exp(2*x)*BesselI(0, 2*x). - Stefano Spezia, May 11 2024

Extensions

Edited by Michael Somos, Sep 13 2003