OFFSET
0,1
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.
FORMULA
G.f.: 2/sqrt(1 - 4*x).
a(n) = 2*A000984(n).
a(n) = 2 * binomial(2*n, n).
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
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
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 *)
PROG
(PARI) a(n)=2*binomial(2*n, n)
(Magma) [2*(n+1)*Catalan(n): n in [0..30]]; // G. C. Greubel, Jul 13 2024
(SageMath) [2*binomial(2*n, n) for n in range(31)] # G. C. Greubel, Jul 13 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by Michael Somos, Sep 13 2003
STATUS
approved