A051960 a(n) = C(n)*(3n+2) where C(n) = Catalan numbers = A000108.

2, 5, 16, 55, 196, 714, 2640, 9867, 37180, 140998, 537472, 2057510, 7904456, 30458900, 117675360, 455657715, 1767883500, 6871173870, 26747767200, 104268528210, 406975466040, 1590307356300, 6220814327520, 24357232569150, 95452906901976, 374369872911804

Offset: 0

Author

Barry E. Williams, Jan 05 2000

Keywords

• easy
• nonn
• changed

Comments

If Y is a fixed 2-subset of a 2n-set X then a(n-1) is the number of n-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007

a(n-1) is the number of vertices in the n-dimensional halohedron (or equivalently, n-cycle cubeahedron). - Vincent Pilaud, May 12 2020

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• Robert Israel, Table of n, a(n) for n = 0..1600
• Moa Apagodu and Doron Zeilberger, Using the "Freshman's Dream" to Prove Combinatorial Congruences, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.
• Satyan L. Devadoss, Timothy Heath, and Cid Vipismakul, Deformations of bordered Riemann surfaces and associahedral polytopes, arXiv:1002.1676 [math.AG], 2010.
• S. L. Devadoss, T. Heath, and W. Vipismakul, Deformations of bordered surfaces and convex polytopes, Notices Amer. Math. Soc. 58 (2011), no. 4, 530-541.
• S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
• Milan Janjic, Two Enumerative Functions.
• Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Crossrefs

Cf. A000108 and A051924.

Cf. A024482 and A097613.

Half A028283.

Programs

Magma

[Catalan(n)*(3*n+2): n in [0..30]]; // Vincenzo Librandi, Oct 01 2015

Maple

a := n -> 4^n*(2+3*n)*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(2+n)):
seq(a(n), n=0..25); # Peter Luschny, Dec 14 2015

Mathematica

Table[CatalanNumber[n] (3n+2), {n,0,30}] (* Michael De Vlieger, Sep 30 2015 *)

Maxima

a(n):=sum(binomial(n-k+1,k)*2^(n-2*k+1)*binomial(n,k),k,0,(n+1)/2); /* Vladimir Kruchinin, Sep 30 2015 */

PARI

a(n) = (3*n+2)*binomial(2*n, n)/(n+1);
vector(30, n, a(n-1)) \\ Altug Alkan, Sep 30 2015

Formula

(n+1)*a(n) - 2*(n+2)*a(n-1) - 4*(2*n-3)*a(n-2) = 0. - conjectured by R. J. Mathar, Oct 02 2014, verified by Robert Israel, Sep 30 2015

G.f.: (1 + 2*x)/(2*x*sqrt(1-4*x)) - 1/(2*x). - Vladimir Kruchinin, Sep 30 2015.

a(n) = Sum_{k=0..(n+1)/2} (binomial(n-k+1,k)*2^(n-2*k+1)*binomial(n,k)). - Vladimir Kruchinin, Sep 30 2015.

a(n) = 4^n*(2+3*n)*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n+2)). - Peter Luschny, Dec 14 2015

a(n - 1) = A051924(n) + A000108(n - 1). - F. Chapoton, Mar 05 2022

Sum_{n>=0} a(n)/8^n = 5*sqrt(2) - 4. - Amiram Eldar, May 06 2023

E.g.f.: exp(2*x)*(2*BesselI(0,2*x) + BesselI(1,2*x)). - Stefano Spezia, May 14 2025

a(n) = 2*binomial(2*n, n) + binomial(2*n, n-1) = 2*A000984(n) + A001791(n). - Peter Bala, Aug 23 2025