cp's OEIS Frontend

A002694 Binomial coefficients C(2n, n-2).

%I A002694 M4181 N1741 #165 Aug 24 2025 04:14:19
%S A002694 1,6,28,120,495,2002,8008,31824,125970,497420,1961256,7726160,
%T A002694 30421755,119759850,471435600,1855967520,7307872110,28781143380,
%U A002694 113380261800,446775310800,1761039350070,6943526580276,27385657281648,108043253365600
%N A002694 Binomial coefficients C(2n, n-2).
%C A002694 Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=2. Example: For n=3 there are 6 paths EEENNN, EENENN, EENNEN, EENNNE, ENEENN and NEEENN. - _Herbert Kociemba_, May 23 2004
%C A002694 Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-2 of which are triangular. Example: a(3)=6 because the convex hexagon ABCDEF is dissected by any of the diagonals AC, BD, CE, DF, EA, FB into regions containing exactly 1 triangle. - _Emeric Deutsch_, May 31 2004
%C A002694 Number of UUU's (triple rises), where U=(1,1), in all Dyck paths of semilength n+1. Example: a(3)=6 because we have UD(UUU)DDD, (UUU)DDDUD, (UUU)DUDDD, (UUU)DDUDD and (U[UU)U]DDDD, the triple rises being shown between parentheses. - _Emeric Deutsch_, Jun 03 2004
%C A002694 Inverse binomial transform of A026389. - _Ross La Haye_, Mar 05 2005
%C A002694 Sum of the jump-lengths of all full binary trees with n internal nodes. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given full binary tree is called the jump-length. - _Emeric Deutsch_, Jan 18 2007
%C A002694 a(n) = number of convex polyominoes (A005436) of perimeter 2n+4 that are directed but not parallelogram polyominoes, because the directed convex polyominoes are counted by the central binomial coefficient binomial(2n,n) and the subset of parallelogram polyominoes is counted by the Catalan number C(n+1) = binomial(2n+2,n+1)/(n+2) and a(n) = binomial(2n,n) - C(n+1). - _David Callan_, Nov 29 2007
%C A002694 a(n) = number of DUU's in all Dyck paths of semilength n+1. Example: a(3)=6 because we have UU(DUU)DDD, U(DUU)UDDD, U(DUU)DUDD, UDU(DUU)DD, U(DUU)DDUD, UUD(DUU)DD, the DUU's being shown between parentheses and no other Dyck path of semilength 4 contains a DUU. - _David Callan_, Jul 25 2008
%C A002694 C(2n,n-m) is the number of Dyck-type walks such that their graphs have one marked edge passed 2m times and the other edges are passed 2 times counting "there and back" directions. - _Oleksiy Khorunzhiy_, Jan 09 2015
%C A002694 Number of paths in the half-plane x >= 0, from (0,0) to (2n,4), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=3, we have the 6 paths: UUUUUD, UUUUDU, UUUDUU, UUDUUU, UDUUUU, DUUUUU, DUUUUU. - _José Luis Ramírez Ramírez_, Apr 19 2015
%D A002694 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D A002694 C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.
%D A002694 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002694 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002694 T. D. Noe, <a href="/A002694/b002694.txt">Table of n, a(n) for n = 2..200</a>
%H A002694 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A002694 Henry Bottomley, <a href="/A002694/a002694.gif">Illustration for A000108, A001147, A002694, A067310 and A067311</a>
%H A002694 Milan Janjic, <a href="https://web.archive.org/web/20181001110015/https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A002694 Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
%H A002694 Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014), Article 14.3.5.
%H A002694 O. Khorunzhiy, <a href="http://arxiv.org/abs/1107.5724">On high moments and the spectral norm of large dilute Wigner random matrices</a>, arXiv:1107.5724 [math-ph], 2014.
%H A002694 O. Khorunzhiy, <a href="https://doi.org/10.15407/mag10.01.064">On high moments and the spectral norm of large dilute Wigner random matrices</a>, Zh. Mat. Fiz. Anal. Geom. 10 (1) (2014), pp. 64-125.
%H A002694 W. Krandick, <a href="http://dx.doi.org/10.1016/j.cam.2003.08.018">Trees and jumps and real roots</a>, J. Computational and Applied Math., 162, 2004, 51-55.
%H A002694 C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)
%H A002694 Asamoah Nkwanta and Earl R. Barnes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Nkwanta/nkwanta2.html">Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind</a>, Journal of Integer Sequences, Article 12.3.3, 2012. - From _N. J. A. Sloane_, Sep 16 2012
%H A002694 V. Pilaud and J. Rué, <a href="http://arxiv.org/abs/1307.6440">Analytic combinatorics of chord and hyperchord diagrams with k crossings</a>, arXiv:1307.6440 [math.CO], 2013; Adv. Appl. Math. 57 (2014) 60-100.
%H A002694 Mark Shattuck, <a href="https://arxiv.org/abs/2303.06300">Enumeration of non-crossing partitions according to subwords with repeated letters</a>, arXiv:2303.06300 [math.CO], 2023.
%H A002694 Daniel W. Stasiuk, <a href="http://hdl.handle.net/10388/11865">An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads</a>, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
%H A002694 T. Tao and Van Vu, <a href="https://arxiv.org/abs/1005.2901">Random matrices: localization of the eigenvalues and the necessity of four moments</a>, arXiv:1005.2901 [math.PR], 2010-2011; Acta Math. Vietnam 36 (2) (2011) 431-449.
%H A002694 N. J. Wildberger and Dean Rubine, <a href="https://doi.org/10.1080/00029890.2025.2460966">A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode</a>, Amer. Math. Monthly (2025). See section 12.
%H A002694 Lin Yang and Shengliang Yang, <a href="https://doi.org/10.4208/jms.v56n1.23.01">Protected Branches in Ordered Trees</a>, J. Math. Study (2023) Vol. 56, No. 1, 1-17.
%F A002694 a(n) = A067310(n, 1) as this is number of ways of arranging n chords on a circle (handshakes between 2n people across a table) with exactly 1 simple intersection. - _Henry Bottomley_, Oct 07 2002
%F A002694 E.g.f.: exp(2*x) * BesselI(2, 2*x). - _Vladeta Jovovic_, Aug 21 2003
%F A002694 G.f.: (1-sqrt(1-4*z))^4/(16*z^2*sqrt(1-4*z)). - _Emeric Deutsch_, Jan 28 2004
%F A002694 a(n) = Sum_{k=0..n} C(n, k)*C(n, k+2). - _Paul Barry_, Sep 20 2004
%F A002694 D-finite with recurrence: -(n-2)*(n+2)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Dec 04 2012
%F A002694 G.f.: z^2*C(z)^4/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - _José Luis Ramírez Ramírez_, Apr 19 2015
%F A002694 a(n) = Sum_{k=1..n} binomial(2*n-k,n-k-1). - _Vladimir Kruchinin_, Oct 22 2016
%F A002694 G.f.: x^2* 2F1(5/2,3;5;4*x). - _R. J. Mathar_, Jan 27 2020
%F A002694 From _Amiram Eldar_, May 16 2022: (Start)
%F A002694 Sum_{n>=2} 1/a(n) = 23/6 - 13*Pi/(9*sqrt(3)).
%F A002694 Sum_{n>=2} (-1)^n/a(n) = 106*log(phi)/(5*sqrt(5)) - 37/10, where phi is the golden ratio (A001622). (End)
%F A002694 From _Peter Bala_, Oct 13 2024: (Start)
%F A002694 a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * (x^2 - 4*x + 2)/sqrt(x*(4 - x)).
%F A002694 G.f. x^2 * B(x) * C(x)^4, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%p A002694 a:=n->sum(binomial(n,j-1)*binomial(n,j+1),j=1..n): seq(a(n), n=2..25); # _Zerinvary Lajos_, Nov 26 2006
%t A002694 CoefficientList[ Series[ 16/(((Sqrt[1 - 4 x] + 1)^4)*Sqrt[1 - 4 x]), {x, 0, 23}], x] (* _Robert G. Wilson v_, Aug 08 2011 *)
%t A002694 Table[Binomial[2n,n-2],{n,2,30}] (* _Harvey P. Dale_, Jun 12 2014 *)
%o A002694 (Haskell)
%o A002694 a002694 n = a007318' (2 * n) (n - 2)  -- _Reinhard Zumkeller_, Jun 18 2012
%o A002694 (Magma) [Binomial(2*n, n-2): n in [2..30]]; // _Vincenzo Librandi_, Apr 20 2015
%o A002694 (PARI) {a(n) = binomial(2*n,n-2)}; \\ _G. C. Greubel_, Mar 21 2019
%o A002694 (Sage) [binomial(2*n,n-2) for n in (2..30)] # _G. C. Greubel_, Mar 21 2019
%o A002694 (GAP) List([2..30], n-> Binomial(2*n,n-2)); # _G. C. Greubel_, Mar 21 2019
%Y A002694 Cf. A006659.
%Y A002694 Diagonal 5 of triangle A100257.
%Y A002694 Cf. A001622, A009766.
%Y A002694 Cf. binomial(k*n, n-k): A000027 (k=1), this sequence (k=2), A004321 (k=3), A004334 (k=4), A004347 (k=5), A004361 (k=6), A004375 (k=7), A004389 (k=8), A281580 (k=9).
%Y A002694 Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.
%K A002694 nonn,easy,changed
%O A002694 2,2
%A A002694 _N. J. A. Sloane_