A051960 - OEIS

%I #75 Nov 21 2023 17:46:06

%S 2,5,16,55,196,714,2640,9867,37180,140998,537472,2057510,7904456,

%T 30458900,117675360,455657715,1767883500,6871173870,26747767200,

%U 104268528210,406975466040,1590307356300,6220814327520,24357232569150,95452906901976,374369872911804

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

%C 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

%C a(n-1) is the number of vertices in the n-dimensional halohedron (or equivalently, n-cycle cubeahedron). - _Vincent Pilaud_, May 12 2020

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

%H Robert Israel, <a href="/A051960/b051960.txt">Table of n, a(n) for n = 0..1600</a>

%H Moa Apagodu and Doron Zeilberger, <a href="http://arxiv.org/abs/1606.03351">Using the "Freshman's Dream" to Prove Combinatorial Congruences</a>, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.

%H Satyan L. Devadoss, Timothy Heath, and Cid Vipismakul, <a href="https://arxiv.org/abs/1002.1676">Deformations of bordered Riemann surfaces and associahedral polytopes</a>, arXiv:1002.1676 [math.AG], 2010.

%H S. L. Devadoss, T. Heath, and W. Vipismakul, <a href="https://www.ams.org/notices/201104/rtx110400530p.pdf">Deformations of bordered surfaces and convex polytopes</a>, Notices Amer. Math. Soc. 58 (2011), no. 4, 530-541.

%H S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.

%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.

%F (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

%F G.f.: (1 + 2*x)/(2*x*sqrt(1-4*x)) - 1/(2*x). - _Vladimir Kruchinin_, Sep 30 2015.

%F 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.

%F a(n) = 4^n*(2+3*n)*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n+2)). - _Peter Luschny_, Dec 14 2015

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

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

%p a := n -> 4^n*(2+3*n)*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(2+n)):

%p seq(a(n), n=0..25); # _Peter Luschny_, Dec 14 2015

%t Table[CatalanNumber[n] (3n+2), {n,0,30}] (* _Michael De Vlieger_, Sep 30 2015 *)

%o (Maxima)

%o 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 */

%o (PARI) a(n) = (3*n+2)*binomial(2*n, n)/(n+1);

%o vector(30, n, a(n-1)) \\ _Altug Alkan_, Sep 30 2015

%o (Magma) [Catalan(n)*(3*n+2): n in [0..30]]; // _Vincenzo Librandi_, Oct 01 2015

%Y Cf. A000108 and A051924.

%Y Cf. A024482 and A097613.

%Y Half A028283.

%K easy,nonn

%O 0,1

%A _Barry E. Williams_, Jan 05 2000