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

%I A051960 #84 Aug 27 2025 01:00:38
%S A051960 2,5,16,55,196,714,2640,9867,37180,140998,537472,2057510,7904456,
%T A051960 30458900,117675360,455657715,1767883500,6871173870,26747767200,
%U A051960 104268528210,406975466040,1590307356300,6220814327520,24357232569150,95452906901976,374369872911804
%N A051960 a(n) = C(n)*(3n+2) where C(n) = Catalan numbers = A000108.
%C A051960 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 A051960 a(n-1) is the number of vertices in the n-dimensional halohedron (or equivalently, n-cycle cubeahedron). - _Vincent Pilaud_, May 12 2020
%D A051960 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A051960 Robert Israel, <a href="/A051960/b051960.txt">Table of n, a(n) for n = 0..1600</a>
%H A051960 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 A051960 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 A051960 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 A051960 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 A051960 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A051960 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%F A051960 (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 A051960 G.f.: (1 + 2*x)/(2*x*sqrt(1-4*x)) - 1/(2*x). - _Vladimir Kruchinin_, Sep 30 2015.
%F A051960 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 A051960 a(n) = 4^n*(2+3*n)*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n+2)). - _Peter Luschny_, Dec 14 2015
%F A051960 a(n - 1) = A051924(n) + A000108(n - 1). - _F. Chapoton_, Mar 05 2022
%F A051960 Sum_{n>=0} a(n)/8^n = 5*sqrt(2) - 4. - _Amiram Eldar_, May 06 2023
%F A051960 E.g.f.: exp(2*x)*(2*BesselI(0,2*x) + BesselI(1,2*x)). - _Stefano Spezia_, May 14 2025
%F A051960 a(n) = 2*binomial(2*n, n) + binomial(2*n, n-1) = 2*A000984(n) + A001791(n). - _Peter Bala_, Aug 23 2025
%p A051960 a := n -> 4^n*(2+3*n)*GAMMA(1/2+n)/(sqrt(Pi)*GAMMA(2+n)):
%p A051960 seq(a(n), n=0..25); # _Peter Luschny_, Dec 14 2015
%t A051960 Table[CatalanNumber[n] (3n+2), {n,0,30}] (* _Michael De Vlieger_, Sep 30 2015 *)
%o A051960 (Maxima)
%o A051960 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 A051960 (PARI) a(n) = (3*n+2)*binomial(2*n, n)/(n+1);
%o A051960 vector(30, n, a(n-1)) \\ _Altug Alkan_, Sep 30 2015
%o A051960 (Magma) [Catalan(n)*(3*n+2): n in [0..30]]; // _Vincenzo Librandi_, Oct 01 2015
%Y A051960 Cf. A000108 and A051924.
%Y A051960 Cf. A024482 and A097613.
%Y A051960 Half A028283.
%K A051960 easy,nonn,changed
%O A051960 0,1
%A A051960 _Barry E. Williams_, Jan 05 2000