The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A051960 a(n) = C(n)*(3n+2) where C(n) = Catalan numbers = A000108. 8
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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 A. 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, 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. Milan Janjic, Two Enumerative Functions Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019. 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 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 *) PROG (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 (MAGMA) [Catalan(n)*(3*n+2): n in [0..30]]; // Vincenzo Librandi, Oct 01 2015 CROSSREFS Cf. A000108. Half A028283. Sequence in context: A321470 A149968 A149969 * A149970 A157418 A149971 Adjacent sequences:  A051957 A051958 A051959 * A051961 A051962 A051963 KEYWORD easy,nonn AUTHOR Barry E. Williams, Jan 05 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 11 15:53 EDT 2021. Contains 343792 sequences. (Running on oeis4.)