

A002694


Binomial coefficients C(2n,n2).
(Formerly M4181 N1741)


31



1, 6, 28, 120, 495, 2002, 8008, 31824, 125970, 497420, 1961256, 7726160, 30421755, 119759850, 471435600, 1855967520, 7307872110, 28781143380, 113380261800, 446775310800, 1761039350070, 6943526580276, 27385657281648, 108043253365600
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OFFSET

2,2


COMMENTS

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 xy=2. Example: For n=3 there are 6 paths EEENNN, EENENN, EENNEN, EENNNE, ENEENN and NEEENN.  Herbert Kociemba, May 23 2004
Number of dissections of a convex (n+3)gon by noncrossing diagonals into several regions, exactly n2 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
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
Inverse binomial transform of A026389.  Ross La Haye, Mar 05 2005
Sum of the jumplengths 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 jumplength.  Emeric Deutsch, Jan 18 2007
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 binom(2n,n) and the subset of parallelogram polyominoes is counted by the Catalan number C(n+1) = binom(2n+2,n+1)/(n+2) and a(n) = binom(2n,n)  C(n+1).  David Callan, Nov 29 2007
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(2n,nm) is the number of Dycktype 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
Number of paths in the halfplane 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


REFERENCES

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.
C. Lanczos, Applied Analysis. PrenticeHall, Englewood Cliffs, NJ, 1956, p. 517.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n=2..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
Milan Janjic, Two Enumerative Functions
M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.  N. J. A. Sloane, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
O. Khorunzhiy, On high moments and the spectral norm of large dilute Wigner random matrices, arXiv:1107.5724 [mathph], 2014.
O. Khorunzhiy, On high moments and the spectral norm of large dilute Wigner random matrices, Zh. Mat. Fiz. Anal. Geom. 10 (1) (2014), pp. 64125.
W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 5155.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Asamoah Nkwanta and Earl R. Barnes, Two Catalantype Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012.  From N. J. A. Sloane, Sep 16 2012
V. Pilaud, J. Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv:1307.6440 [math.CO], 2013; Adv. Appl. Math. 57 (2014) 60100.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of nary Trees Arising from Algebraic Operads, Master's Thesis, University of SaskatchewanSaskatoon (2018).
T. Tao, Van Vu, Random matrices: localization of the eigenvalues and the necessity of four moments, arXiv:1005.2901 [math.PR]20102011; Acta Math. Vietnam 36 (2) (2011) 431449.


FORMULA

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
E.g.f.: exp(2*x)*BesselI(2, 2*x).  Vladeta Jovovic, Aug 21 2003
G.f.: (1sqrt(14*z))^4/(16*z^2*sqrt(14*z)).  Emeric Deutsch, Jan 28 2004
a(n) = Sum_{k=0..n} C(n, k)*C(n, k+2).  Paul Barry, Sep 20 2004
(n2)*(n+2)*a(n) +2*n*(2*n1)*a(n1)=0.  R. J. Mathar, Dec 04 2012
G.f.: z^2*C(z)^4/(12*z*C(z)), where C(z) is the g.f. of Catalan numbers.  José Luis Ramírez Ramírez, Apr 19 2015
a(n) = Sum_{k=1..n} binomial(2*nk,nk1).  Vladimir Kruchinin, Oct 22 2016


MAPLE

a:=n>sum(binomial(n, j1)*binomial(n, j+1), j=1..n): seq(a(n), n=2..25); # Zerinvary Lajos, Nov 26 2006


MATHEMATICA

CoefficientList[ Series[ 16/(((Sqrt[1  4 x] + 1)^4)*Sqrt[1  4 x]), {x, 0, 23}], x] (* Robert G. Wilson v, Aug 08 2011 *)
Table[Binomial[2n, n2], {n, 2, 30}] (* Harvey P. Dale, Jun 12 2014 *)


PROG

(Haskell)
a002694 n = a007318' (2 * n) (n  2)  Reinhard Zumkeller, Jun 18 2012
(MAGMA) [Binomial(2*n, n2): n in [2..30]]; // Vincenzo Librandi, Apr 20 2015
(PARI) {a(n) = binomial(2*n, n2)}; \\ G. C. Greubel, Mar 21 2019
(Sage) [binomial(2*n, n2) for n in (2..30)] # G. C. Greubel, Mar 21 2019
(GAP) List([2..30], n> Binomial(2*n, n2)) # G. C. Greubel, Mar 21 2019


CROSSREFS

Cf. A006659.
Diagonal 5 of triangle A100257.
Cf. A009766.
Cf. binomial(k*n, nk): 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).
Sequence in context: A090777 A055715 A026031 * A007691 A260508 A065997
Adjacent sequences: A002691 A002692 A002693 * A002695 A002696 A002697


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



