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A000108
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Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
(Formerly M1459 N0577)
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2023
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1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304
(list; graph; refs; listen; history; internal format)
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OFFSET
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COMMENTS
| The solution to Schroeder's first problem. A very large number of combinatorial interpretations are known - see references, esp. Stanley, Enumerative Combinatorics, Volume 2.
Number of ways to insert n pairs of parentheses in a word of n+1 letters. E.g. for n=3 there are 5 ways: ((ab)(cd)), (((ab)c)d), ((a(bc))d), (a((bc)d)), (a(b(cd))).
Consider all the binomial(2n,n) paths on squared paper that (i) start at (0, 0), (ii) end at (2n, 0) and (iii) at each step, either make a (+1,+1) step or a (+1,-1) step. Then the number of such paths which never go never below the x-axis (Dyck paths) is C(n) [Chung-Feller]
Number of noncrossing partitions of the n-set. For example, of the 15 set partitions of the 4-set, only [{13},{24}] is crossing, so there are a(4)=14 noncrossing partitions of 4 elements. [Joerg Arndt, Jul 11 2011]
a(n-1) is the number of ways of expressing an n-cycle in the symmetric group S_n as a product of n-1 transpositions (u_1,v_1)*(u_2,v_2)*...*(u_{n-1},v_{n-1}) where uk<=uj and vk<=vj for k<j; see example. If the condition is dropped one obtains A000272. [Joerg Arndt and Greg Stevenson, Jul 11 2011]
a(n) is the number of ordered rooted trees with n nodes, not including the root. See the Conway-Guy reference where these rooted ordered trees are called plane bushes. See also the Bergeron et al. reference, Example 4, p. 167. W. Lang Aug 07 2007.
Shifts one place left when convolved with itself.
For n >= 1 a(n) is also the number of rooted bicolored unicellular maps of genus 0 on n edges. - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 15 2001
Ways of joining 2n points on a circle to form n nonintersecting chords. (If no such restriction imposed, then ways of forming n chords is given by (2n-1)!!=(2n)!/n!2^n=A001147(n).)
Arises in Schubert calculus - see Sottile reference.
Inverse Euler transform of sequence is A022553.
With interpolated zeros, the inverse binomial transform of the Motzkin numbers A001006. - Paul Barry, Jul 18 2003
The Hankel transforms of this sequence or of this sequence with the first term omitted give A000012 = 1, 1, 1, 1, 1, 1, ...; example : Det([1, 1, 2, 5; 1, 2, 5, 14; 2, 5, 14, 42; 5, 14, 42, 132]) = 1 and Det([1, 2, 5, 14; 2, 5, 14, 42; 5, 14, 42, 132; 14, 42, 132, 429]) = 1 . - DELEHAM Philippe, Mar 04 2004
c(n) = C(2*n-2,n-1)/n = (1/n!) * [ n^(n-1) + { C(n-2,1) +C(n-2,2) }*n^(n-2) + { 2*C(n-3,1) +7*C(n-3,2) +8*C(n-3,3) +3*C(n-3,4) }*n^(n-3) + { 6*C(n-4,1) +38*C(n-4,2) +93*C(n-4,3) +111*C(n-4,4) +65*C(n-4,5) +15*C(n-4,6) }*n^(n-4) + ..... ]. - Andre F. Labossiere (boronali(AT)laposte.net), Nov 10 2004
Sum_{n=0..infinity} 1/a(n) = 2 + 4*Pi/3^(5/2) = F(1,2;1/2;1/4) = 2.806133050770763... (see L'Universe de Pi link) - Gerald McGarvey and Benoit Cloitre, Feb 13 2005
a(n) equals sum of squares of terms in row n of triangle A053121, which is formed from successive self-convolutions of the Catalan sequence. - Paul D. Hanna, Apr 23 2005
Comment from Donald D. Cross (cosinekitty(AT)hotmail.com), Feb 04 2005: Also coefficients of the Mandelbrot polynomial M iterated an infinite number of times. Examples: M(0) = 0 = 0*c^0 = [0], M(1) = c = c^1 + 0*c^0 = [1 0], M(2) = c^2 + c = c^2 + c^1 + 0*c^0 = [1 1 0], M(3) = (c^2 + c)^2 + c = [0 1 1 2 1], ... ... M(5) = [0 1 1 2 5 14 26 44 69 94 114 116 94 60 28 8 1], ...
The multiplicity with which a prime p divides C_n can be determined by first expressing n+1 in base p. For p=2, the multiplicity is the number of 1 digits minus 1. For p an odd prime, count all digits greater than (p+1)/2; also count digits equal to (p+1)/2 unless final; and count digits equal to (p-1)/2 if not final and the next digit is counted. For example, n=62, n+1 = 223_5, so C_62 is not divisible by 5. n=63, n+1 = 224_5, so 5^3 | C_63. - Frank Adams-Watters, Feb 08 2006
Koshy and Salmassi give an elementary proof that the only prime Catalan numbers are a(2) = 2 and a(3) = 5. Is the only semiprime Catalan number a(4) = 14? - Jonathan Vos Post, Mar 06 2006
Comment from Franklin T. Adams-Watters, Apr 14 2006: The answer is yes. Using the formula C_n = C(2n,n)/(n+1), it is immediately clear that C_n can have no prime factor greater than 2n. For n >= 7, C_n > (2n)^2, so it cannot be a semiprime. Given that the Catalan numbers grow exponentially, the above consideration implies that the number of prime divisors of C_n, counted with multiplicity, must grow without limit. The number of distinct prime divisors must also grow without limit, but this is more difficult. Any prime between n+1 and 2n (exclusive) must divide C_n. That the number of such primes grows without limit follows from the prime number theorem.
The number of ways to place n indistinguishable balls in n numbered boxes B1,...,Bn such that at most a total of k balls are placed in boxes B1,...,Bk for k=1,...,n. For example, a(3)=5 since there are 5 ways to distribute 3 balls among 3 boxes such that (i) box 1 gets at most 1 ball and (ii)box 1 and box 2 together get at most 2 balls:(O)(O)(O), (O)()(OO), ()(OO)(O), ()(O)(OO), ()()(OOO). - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Dec 04 2006
a(n) is also the order of the semigroup of order-decreasing and order-preserving full transformations (of an n-element chain) - now known as the Catalan monoid [From A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008]
a(n) is the number of trivial representations in the direct product of 2n spinor (the smallest) representations of the group SU(2) (A(1)). [From Rutger Boels (boels(AT)nbi.dk), Aug 26 2008]
The invert transform appears to converge to the catalan numbers when applied infinitely many times to any starting sequence. [From Mats O. Granvik, Gary W. Adamson and Roger L. Bagula, Sep 09 2008, Sep 12 2008]
lim(a(n)/a(n-1): n->infinity) = 4 [From Francesco Antoni (francesco_antoni(AT)yahoo.com), Nov 24 2008]
Starting with offset 1 = row sums of triangle A154559 [From Gary W. Adamson, Jan 11 2009]
C(n) is the degree of the Grassmanian G(1,n+1): the set of lines in (n+1)-dimensional projective space, or the set of planes through the origin in (n+2)-dimensional affine space. The Grassmanian is considered a subset of N-dimensional projective space, N = binomial(n+2,2) - 1. If we choose 2n general (n-1)-planes in projective (n+1)-space, then there are C(n) lines that meet all of them. [Benji Fisher (benji(AT)FisherFam.org), Mar 05 2009]
Contribution from Gary W. Adamson, May 01 2009: (Start)
Starting with offset 1 = A068875: (1, 2, 4, 10, 18, 84,...) convolved with
Fine numbers, A000957: (1, 0, 1, 2, 6, 18,...). a(6) = 132 =
(1, 2, 4, 10, 28, 84) dot (18, 6, 2, 1, 0, 1) = (18 + 12 + 8 + 10 + 0 + 84) = 132. (End)
Convolved with A032443: (1, 3, 11, 42, 163,...) = powers of 4, A000302: (1, 4, 16,...). [From Gary W. Adamson, May 15 2009]
Sum{k=1...Infinity,c(k-1)/2^(2k-1)}=1. The k-th term in the summation is the probability that a random walk on the integers (begining at the origin) will arrive at positive one (for the first time) in exactly (2k-1) steps. [From Geoffrey Critzer, Sep 12 2009]
C(p+q)-C(p)*C(q)=sum(C(i)*C(j)*C(p+q-i-j-1), i=0..(p-1), j=0..(q-1) ) [From Groux roland, Nov 13 2009]
Leonhard Euler used the formula C(n) = product_{i=3..n}(4*i-10)/(i-1) in his 'Betrachtungen, auf wie vielerley Arten ein gegebenes polygonum durch Diagonallinien in triangula zerschnitten werden k"onne' and computes by recursion C(n+2) for n = 1..8. (Berlin, 4th September 1751, in a letter to Goldbach). [From Peter Luschny, Mar 13 2010]
Let A179277 = A(x). Then C(x) is satisfied by A(x)/A(x^2). [From Gary W. Adamson, Jul 07 2010]
a(n)= A000680(n)/A006472(n) [From M.dols (markdols99(AT)yahoo.com), Jul 14 2010]
a(n) is also the number of quivers in the mutation class of type B_n or of type C_n. [From Christian Stump (christian.stump(AT)gmail.com), Nov 02 2010]
Consider a set of A000217(n) balls of n colors in which, for each integer k = 1 to n, exactly one color appears in the set a total of k times. (Each ball has exactly one color and is indistinguishable from other balls of the same color.) a(n+1) equals the number of ways to choose 0 or more balls of each color while satisfying the following conditions: 1. No two colors are chosen the same positive number of times. 2. For any two colors (c, d) that are chosen at least once, color c is chosen more times than color d iff color c appears more times in the original set than color d.
If the second requirement is lifted, the number of acceptable ways equals A000110(n+1). See related comments for A016098, A085082. [From Matthew Vandermast, Nov 22 2010]
Deutsch and Sagan prove the Catalan number C_n is odd if and only if n = 2^a - 1 for some nonnegative integer a. Lin proves for every odd Catalan number C_n, we have C_n == 1 (mod 4). [Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 09 2010]
a(n) is the number of functions f:{1,2,...,n}->{1,2,...,n} such that f(1)=1 and for all n>=1 f(n+1)<=f(n)+1. For a nice bijection between this set of functions and the set of length 2n Dyck words see page 333 of the fxtbook (see link below).
Complement of A092459; A010058(a(n)) = 1. [Reinhard Zumkeller, Mar 29 2011]
Postnikov (2005) defines "generalized Catalan numbers" associated with buildings (e.g. Catalan numbers of Type B, see A000984). - N. J. A. Sloane, Dec 10 2011.
A076050(a(n)) = n + 1 for n > 0. [Reinhard Zumkeller, Feb 17 2012]
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REFERENCES
| The large number of references and links demonstrates the ubiquity of the Catalan numbers.
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Wen-jin Woan, A Relation Between Restricted and Unrestricted Weighted Motzkin Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.7.
Wen-jin Woan, Animals and 2-Motzkin Paths, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.6.
F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.
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LINKS
| N. J. A. Sloane, The first 200 Catalan numbers
Joerg Arndt, Fxtbook, p.333 and p.337.
Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48. (C(n) = number of triangulations of cyclic polytope C(n,2).)
John Baez, This week's finds in mathematical physics, Week 202
E. Barcucci, A. Del Lungo, E. Pergola and R. Pinzani, Permutations avoiding an increasing number of length-increasing forbidden subsequences
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490, 2011.
Matthew Bennett, Vyjayanthi Chari, R. J. Dolbin and Nathan Manning, Square partitions and Catalan numbers, arXiv:0912.4983.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
D. Bill, Durango Bill's Enumeration of Binary Trees
H. Bottomley, Catalan Space Invaders
H. Bottomley, Illustration for A000108, A001147, A002694, A067310 and A067311
T. Bourgeron, Montagnards et polygones
M. Bousquet-Melou and Gilles Schaeffer, Walks on the slit plane, Probability Theory and Related Fields, Vol. 124, no. 3 (2002), 305-344.
K. S. Brown's Mathpages, The Meanings of Catalan Numbers
B. Bukh, PlanetMath.org, Catalan numbers
Alexander Burstein, Sergi Elizalde and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv math.CO/0610234.
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
Julie Christophe, Jean-Paul Doignon and Samuel Fiorini, Counting Biorders, J. Integer Seqs., Vol. 6, 2003.
J. Cigler, Some nice Hankel determinants, arXiv:1109.1449, 2011.
T. Davis, Catalan Numbers
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
R. M. Dickau, Catalan numbers (another copy)
Philippe Flajolet, Eric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics, arXiv:math.CO/0606370
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 18, 35
D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence
I. Galkin, Enumeration of the Binary Trees(Catalan Numbers)
Mohammad GANJTABESH, Armin MORABBI and Jean-Marc STEYAERT, Enumerating the number of RNA structures
H. W. Gould, Congr. Numer. 165 (2003) p 33-38.
B. Gourevitch, L'univers de Pi (click Mathematiciens, Gosper)
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)
Guo-Niu Han, Enumeration of Standard Puzzles
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 48
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 52
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 71
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 76
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 284
I. Jensen, Series exapansions for self-avoiding polygons
S. Johnson, The Catalan Numbers
A. Karttunen, Illustration of initial terms up to size n=7
C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
André F. Labossière, Coefficients Binomiaux des Diagonales du Triangle de Pascal.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Hsueh-Yung Lin, The odd Catalan numbers modulo 2^k, Dec 08 2010.
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Toufik Mansour, Counting Peaks at Height k in a Dyck Path, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.1
Toufik Mansour and Yidong Sun, Identities involving Narayana polynomials and Catalan numbers (2008), arXiv:0805.1274; Discrete Mathematics, Volume 309, Issue 12, Jun 28 2009, Pages 4079-4088
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J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions of arbitrary level
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A. Panholzer and H. Prodinger, Bijections for ternary trees and non-crossing trees, Discrete Math., 250 (2002), 181-195 (see Eq. 4).
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P. Peart and W.-J. Woan, Dyck Paths With No Peaks at Height k, J. Integer Sequences, 4 (2001), #01.1.3.
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N. J. A. Sloane, Illustration of initial terms
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R. P. Stanley, Exercises on Catalan and Related Numbers
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V. S. Sunder, Catalan numbers
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A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv:1107.2938, 2011.
G. Villemin's Almanac of Numbers, Nombres De Catalan
Eric Weisstein's World of Mathematics, Catalan Number
Eric Weisstein's World of Mathematics, Binary Bracketing
Eric Weisstein's World of Mathematics, Binary Tree
Eric Weisstein's World of Mathematics, Nonassociative Product
Eric Weisstein's World of Mathematics, Staircase Walk
Eric Weisstein's World of Mathematics, Dyck Path
W.-J. Woan, Hankel Matrices and Lattice Paths, J. Integer Sequences, 4 (2001), #01.1.2.
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to parenthesizing
Index entries for sequences related to necklaces
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FORMULA
| a(n) = binomial(2n, n)/(n+1) = (2n)!/(n!(n+1)!).
a(n) = binomial(2n, n)-binomial(2n, n-1)
a(n) = Sum_{k=0..n-1} a(k)a(n-1-k).
G.f.: A(x) = (1 - sqrt(1 - 4*x)) / (2*x). G.f. A(x) satisfies A = 1 + x*A^2.
a(n+1) = Sum_{i} binomial(n, 2*i)*2^(n-2*i)*a(i) - Touchard.
2(2n-1)a(n-1) = (n+1)a(n).
It is known that a(n) is odd if and only if n=2^k-1, k=1, 2, 3, ... - Emeric Deutsch, Aug 04 2002.
Using the Stirling approximation in A000142 we get the asymptotic expansion a(n) ~ 4^n / (sqrt(Pi * n) * (n + 1)). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001
Integral representation: a(n)=int(x^n*sqrt((4-x)/x), x=0..4)/(2*Pi). - Karol A. Penson (penson(AT)lptl.jussieu.fr), Apr 12 2001
E.g.f.: exp(2x) (I_0(2x)-I_1(2x)), where I_n is Bessel function. - Karol A. Penson, Oct 07 2001
Polygorial(n, 6)/Polygorial(n, 3) - Daniel Dockery (peritus(AT)gmail.com) Jun 24, 2003
G.f. A(x) satisfies ((A(x)+A(-x))/2)^2 = A(4*x^2). - Michael Somos, Jun 27, 2003
G.f. A(x) satisfies Sum_{k>=1} k(A(x)-1)^k = Sum_{n >= 1} 4^{n-1} x^n. - Shapiro, Woan, Getu
a(n+m) = Sum_{k} A039599(n, k)*A039599(m, k). - DELEHAM Philippe, Dec 22 2003
a(n+1) = (1/(n+1))*sum_{k=0..n} a(n-k)*binomial(2k+1, k+1) . - DELEHAM Philippe, Jan 24 2004
a(n) = Sum_{k>=0} A008313(n, k)^2 . - DELEHAM Philippe, Feb 14 2004
a(m+n+1) = Sum_{k>=0} A039598(m, k)*A039598(n, k) . - DELEHAM Philippe, Feb 15 2004
a(n)=sum{k=0..n, (-1)^k*2^(n-k)*binomial(n, k)*binomial(k, floor(k/2))} - Paul Barry, Jan 27 2005
a(n) = Sum_{k=0..[n/2]} ((n-2*k+1)*C(n, n-k)/(n-k+1))^2, which is equivalent to: a(n) = Sum_{k=0..n} A053121(n, k)^2, for n>=0. - Paul D. Hanna, Apr 23 2005
a((m+n)/2) = Sum_{k>=0} A053121(m, k)*A053121(n, k) if m+n is even . - Philippe DELEHAM, May 26 2005
E.g.f. Sum_{n>=0} a(n)*x^(2n)/(2n)! = BesselI(1, 2x)/x . - Michael Somos, Jun 22 2005
Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(x, B(X)) where f(u, v)=u-v+(uv)^2 or B(x)=x+(x*B(x))^2 which implies B(-B(x))=-x and also (1+B^3)/B^2 = (1-x^3)/x^2 . - Michael Somos Jun 27 2005
a(n) = a(n-1)*(4-6/(n+1)). a(n) = 2a(n-1)*(8a(n-2)+a(n-1))/(10a(n-2)-a(n-1)). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 08 2006
Sum_{k=1}^{infinity} a(k)/4^k = 1. - Frank Adams-Watters, Jun 28 2006
a(n) = A047996(2*n+1,n) . - DELEHAM Philippe, Jul 25 2006
Binomial transform of A005043 . - Philippe DELEHAM, Oct 20 2006
a(n)=Sum_{k, 0<=k<=n}(-1)^k*A116395(n,k) . - Philippe DELEHAM, Nov 07 2006
a(n)=[1/(s-n)]*sum_{k=0..n} (-1)^k (k+s-n)*binomial(s-n,k)*binomial(s+n-k,s) with s a nonnegative free integer [H. W. Gould].
a(k) = Sum_{i=1..k} |A008276(i,k)| * (k-1)^(k-i) / k! - Andre F. Labossiere (boronali(AT)laposte.net), May 29 2007
a(n)=Sum_{k, 0<=k<=n}A129818(n,k)*A007852(k+1). - Philippe DELEHAM, Jun 20 2007
a(n)=Sum_{k, 0<=k<=n}A109466(n,k)*A127632(k). - Philippe DELEHAM, Jun 20 2007
Row sums of triangle A124926 - Gary W. Adamson, Oct 22 2007
For G.f. A(x), g(x)= x*A(x) is the compositional inverse of f(x) = x*(1-x) and this relates the Catalan numbers to the row sums of A125181. - Tom Copeland, Jan 13 2008
lim(1+Sum(a(k)/A004171(k): 0<=k<=n): n->infinity) = 4/pi. [From Reinhard Zumkeller, Aug 26 2008]
a(n)=Sum_{k, 0<=k<=n}A120730(n,k)^2 and a(k+1)=Sum_{n, n>=k}A120730(n,k). [From Philippe DELEHAM, Oct 18 2008]
Comment from Gary W. Adamson, Oct 27 2008: Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example the present sequence is Phi([1]) (also Phi([1,1])).
Formula from Thomas Wieder, Feb 25 2009:
a(n) = sum_{l_1=0}^{n+1} sum_{l_2=0}^{n}...sum_{l_i=0}^{n-i}...sum_{l_n=0}^{1}
delta(l_1,l_2,...,l_i,...,l_n)
where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i < l_(i+1) and l_(i+1) <> 0
for i=1..n-1 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise.
C(n) = (4 - 6/n) * C(n-1) with C(1) = 1 [From M. Dols (markdols99(AT)yahoo.com), Feb 14 2010]
G.f. A(x), B(x)=x*A(x) satisfies the differential equation B'(x)-2*B'(x)*B(x)-1=0 [From Vladimir Kruchinin, Jan 18 2011]
G.f.: 1/(1-x/(1-x/(1-x/(...)))) (continued fraction). [Joerg Arndt, Mar 18 2011]
Contribution from Tom Copeland, Sept 04 2011: (Start)
With F(x) = (1-2*x-sqrt(1-4*x))/(2*x) an o.g.f. in x for the Catalan series, G(x)= x/(1+x)^2 is the compositional inverse of F (nulling the n=0 term).
With H(x) = 1/(dG(x)/dx) = (1+x)^3 / (1-x), the n-th Catalan number is given by (1/n!)*((H(x)*d/dx)^n)x evaluated at x=0, i.e., F(x) = exp(x*H(u)*d/du)u, evaluated at u = 0. Also, dF(x)/dx = H(F(x)), and H(x) is the o.g.f. for A115291. (End)
Contribution from Tom Copeland, Sept 30 2011: (Start)
With F(x)={1-sqrt[1-4*x]}/2 an o.g.f. in x for the Catalan series,
G(x)= x*(1-x) is the compositional inverse.
With H(x)=1/(dG(x)/dx)= 1/(1-2x), the n-th Catalan number (offset 1) is given by (1/n!)*((H(x)*d/dx)^n)x evaluated at x=0, i.e.,
F(x) = exp(x*H(u)*d/du)u, evaluated at u = 0. Also, dF(x)/dx = H(F(x)).(End)
G.f.: A(x)=(1-sqrt(1-4*x))/(2*x)=G(0);
G(k)=1+(4*k+1)*x/(k+1-2*x*(k+1)*(4*k+3)/(2*x*(4*k+3)+(2*k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2011
E.g.f.: A(x)=exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x))=G(0);
G(k)=1+(4*k+1)*x/((k+1)*(2*k+1)-x*(k+1)*(2*k+1)*(4*k+3)/(x*(4*k+3)+(k+1)*(2*k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2011
E.g.f.: Hypergeometric([1/2],[2],4*x) which coincides with the e.g.f. given just above, and also by Karol A. Penson further above. - Wolfdieter Lang, Jan 13 2012.
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EXAMPLE
| The following products of 3 transpositions lead to a 4-cycle in S_4:
(1,2)*(1,3)*(1,4);
(1,2)*(1,4)*(3,4);
(1,3)*(1,4)*(2,3);
(1,4)*(2,3)*(2,4);
(1,4)*(2,4)*(3,4). [Joerg Arndt and Greg Stevenson, Jul 11 2011]
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MAPLE
| A000108 := n->binomial(2*n, n)/(n+1); G000108 := (1 - sqrt(1 - 4*x)) / (2*x);
spec := [ A, {A=Prod(Z, Sequence(A))}, unlabeled ]: [ seq(combstruct[count](spec, size=n), n=0..42) ];
with(combstruct):bin := {B=Union(Z, Prod(B, B))}: seq (count([B, bin, unlabeled], size=n), n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 05 2007
Z[0]:=0: for k to 42 do Z[k]:=simplify(1/(1-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..42): gser:=series(g, z=0, 42): seq(coeff(gser, z, n), n=0..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 21 2008
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MATHEMATICA
| A000108[ n_ ] := (2 n)!/n!/(n+1)!
A000108[n_] := Hypergeometric2F1[1 - n, -n, 2, 1] (* Richard L. Ollerton, Sep 13 2006 *)
Table[ CatalanNumber@ n, {n, 0, 24}] (* RGWv, Feb 15 2011 *)
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PROG
| (MAGMA) C:= func< n | Binomial(2*n, n)/(n+1) >; [ C(n) : n in [0..60]];
(PARI) a(n)=if(n<0, 0, (2*n)!/n!/(n+1)!)
(PARI) a(n)=local(A, m); if(n<0, 0, m=1; A=1+x+O(x^2); while(m<=n, m*=2; A=sqrt(subst(A, x, 4*x^2)); A+=(A-1)/(2*x*A)); polcoeff(A, n))
(PARI) a(n)=if(n<1, n==0, polcoeff(serreverse(x/(1+x)^2+x*O(x^n)), n)) (from Michael Somos)
(Mupad) combinat::dyckWords::count(n) $ n = 0..38 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 14 2007
(Sage) [catalan_number(i) for i in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008
(Sage) [binomial(2*i, i)-binomial(2*i, i-1) for i in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
(MAGMA) [Catalan(n): n in [0..40]]; - Vincenzo Librandi, Apr 02 2011
(Haskell)
a000108 n = a000108_list !! n
a000108_list = 1 : catalan [1] where
catalan cs = c : catalan (c:cs) where
c = sum $ zipWith (*) cs $ reverse cs
-- Reinhard Zumkeller, Nov 12 2011
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CROSSREFS
| Cf. A000984, A002420, A048990, A024492, A000142, A022553, A039599, A094216, A094638, A014137, A094639, A099731, A008549, A008276, A094638 (|A008276|), A094216, A094639, A000984, A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, A124926, A098597, A086117, A137697.
A row of A060854.
See A001003, A001190, A001699, A000081 for other ways to count parentheses.
Enumerates objects encoded by A014486.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
A diagonal of the square array described in A051168.
Cf. A000957, A068875, A032443, A179277, A154559 [From Gary W. Adamson]
Partitions into Catalan numbers: A033552, A176137. [From Reinhard Zumkeller, Apr 10 2010]
Sequence in context: A168491 A115140 A120588 * A057413 A126567 A125501
Adjacent sequences: A000105 A000106 A000107 * A000109 A000110 A000111
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KEYWORD
| core,nonn,easy,eigen,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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