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A000108 Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
(Formerly M1459 N0577)
2866

%I M1459 N0577

%S 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,

%T 9694845,35357670,129644790,477638700,1767263190,6564120420,

%U 24466267020,91482563640,343059613650,1289904147324,4861946401452,18367353072152,69533550916004,263747951750360,1002242216651368,3814986502092304

%N Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.

%C The solution to Schröder's first problem. A very large number of combinatorial interpretations are known - see references, esp. Stanley, Enumerative Combinatorics, Volume 2. This is probably the longest entry in the OEIS, and rightly so.

%C Number of ways to insert n pairs of parentheses in a word of n+1 letters. E.g., for n=2 there are 2 ways: ((ab)c) or (a(bc)); for n=3 there are 5 ways: ((ab)(cd)), (((ab)c)d), ((a(bc))d), (a((bc)d)), (a(b(cd))).

%C 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 that never go below the x-axis (Dyck paths) is C(n). [Chung-Feller]

%C 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

%C 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 u_k<=u_j and v_k<=v_j for k<j; see example. If the condition is dropped, one obtains A000272. - _Joerg Arndt_ and Greg Stevenson, Jul 11 2011

%C 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. - _Wolfdieter Lang_, Aug 07 2007

%C As shown in the paper from Beineke and Pippert (1971), a(n-2)=D(n) is the number of labeled dissections of a disk, related to the number R(n)=A001761(n-2) of labeled planar 2-trees having n vertices and rooted at a given exterior edge, by the formula D(n)=R(n)/(n-2)!. - _M. F. Hasler_, Feb 22 2012

%C Shifts one place left when convolved with itself.

%C 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

%C 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).)

%C Arises in Schubert calculus - see Sottile reference.

%C Inverse Euler transform of sequence is A022553.

%C With interpolated zeros, the inverse binomial transform of the Motzkin numbers A001006. - _Paul Barry_, Jul 18 2003

%C 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. - _Philippe Deléham_, Mar 04 2004

%C 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

%C 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], ... - Donald D. Cross (cosinekitty(AT)hotmail.com), Feb 04 2005

%C 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. - _Franklin T. Adams-Watters_, Feb 08 2006

%C 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

%C The answer is yes. Using the formula C_n = binomial(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. - _Franklin T. Adams-Watters_, Apr 14 2006

%C 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_, Dec 04 2006

%C 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. - _Abdullahi Umar_, Aug 25 2008

%C 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)). - Rutger Boels (boels(AT)nbi.dk), Aug 26 2008

%C The invert transform appears to converge to the Catalan numbers when applied infinitely many times to any starting sequence. - _Mats Granvik_, _Gary W. Adamson_ and _Roger L. Bagula_, Sep 09 2008, Sep 12 2008

%C lim(a(n)/a(n-1): n->infinity) = 4. - Francesco Antoni (francesco_antoni(AT)yahoo.com), Nov 24 2008

%C Starting with offset 1 = row sums of triangle A154559. - _Gary W. Adamson_, Jan 11 2009

%C C(n) is the degree of the Grassmannian 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 Grassmannian 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

%C 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. - _Gary W. Adamson_, May 01 2009

%C Convolved with A032443: (1, 3, 11, 42, 163, ...) = powers of 4, A000302: (1, 4, 16, ...). - _Gary W. Adamson_, May 15 2009

%C 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. - _Geoffrey Critzer_, Sep 12 2009

%C C(p+q)-C(p)*C(q) = Sum_{i=0..p-1, j=0..q-1} C(i)*C(j)*C(p+q-i-j-1). - _Groux Roland_, Nov 13 2009

%C 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 könne' and computes by recursion C(n+2) for n = 1..8. (Berlin, 4th September 1751, in a letter to Goldbach.) - _Peter Luschny_, Mar 13 2010

%C Let A179277 = A(x). Then C(x) is satisfied by A(x)/A(x^2). - _Gary W. Adamson_, Jul 07 2010

%C a(n) = A000680(n)/A006472(n+1). - _Mark Dols_, Jul 14 2010; corrected by _M. F. Hasler_, Nov 08 2015

%C a(n) is also the number of quivers in the mutation class of type B_n or of type C_n. - _Christian Stump_, Nov 02 2010

%C 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.

%C If the second requirement is lifted, the number of acceptable ways equals A000110(n+1). See related comments for A016098, A085082. - _Matthew Vandermast_, Nov 22 2010

%C 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_, Dec 09 2010

%C 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).

%C Complement of A092459; A010058(a(n)) = 1. - _Reinhard Zumkeller_, Mar 29 2011

%C 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

%C A076050(a(n)) = n + 1 for n > 0. - _Reinhard Zumkeller_, Feb 17 2012

%C Number of permutations in S(n) for which length equals depth. - _Bridget Tenner_, Feb 22 2012

%C a(n) is also the number of standard Young tableau of shape (n,n). - _Thotsaporn Thanatipanonda_, Feb 25 2012

%C a(n) is the number of binary sequences of length 2n+1 in which the number of ones first exceed the number of zeros at entry 2n+1. See the example below in the example section. - _Dennis P. Walsh_, Apr 11 2012

%C a(n+1) = A214292(2*n+1,n) = A214292(2*n+2,n). - _Reinhard Zumkeller_, Jul 12 2012

%C Number of binary necklaces of length 2*n+1 containing n 1's (or, by symmetry, 0's). All these are Lyndon words and their representatives (as cyclic maxima) are the binary Dyck words. - _Joerg Arndt_, Nov 12 1012

%C Number of sequences consisting of n 'x' letters and n 'y' letters such that (counting from the left) the 'x' count >= 'y' count. For example, for n=3 we have xxxyyy, xxyxyy, xxyyxy, xyxxyy and xyxyxy. - _Jon Perry_, Nov 16 2012

%C a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps come in 2 colors. Example: a(4)=14 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 8 paths of shape HHH, 2 paths of shape UHD, 2 paths of shape UDH, and 2 paths of shape HUD. - _José Luis Ramírez Ramírez_, Jan 16 2013

%C If p is an odd prime, then (-1)^((p-1)/2)*a((p-1)/2) mod p = 2. - _Gary Detlefs_, Feb 20 2013

%C Conjecture: For any positive integer n, the polynomial Sum_{k=0..n} a(k)*x^k is irreducible over the field of rational numbers. - _Zhi-Wei Sun_, Mar 23 2013

%C a(n) is the size of the Jones monoid on 2n points (cf. A225798). - _James Mitchell_, Jul 28 2013

%C Given Probability (p): Sum_{n=0..infinity} a(n)*(1-p)^n*p^(n+1) = Sum_{n=1..infinity} p^n = p/(1-p). E.g., at p=0.4: 0.4 + 0.6*0.4^2 + 2*0.6^2*0.4^3 + 5*0.6^3*0.4^4 + 14*0.6^4*0.4^5 +... = 0.4 + 0.096 + 0.04608 + 0.027648 + 0.018579456... = 2/3. Since p/(1-p) is itself a probability, it therefore has a maximum value of 1 when p >= 0.5. - _Bob Selcoe_, Nov 16 2013

%C No a(n) has the form x^m with m > 1 and x > 1. - _Zhi-Wei Sun_, Dec 02 2013

%C From _Alexander Adamchuk_, Dec 27 2013: (Start)

%C Prime p divides a((p+1)/2) for p>3. See A120303(n) = Largest prime factor of Catalan number.

%C Reciprocal Catalan Constant C = 1 + 4*sqrt(3)*Pi/27 = 1.8061330507707... See A121839 = Decimal expansion of Sum_{k>=1} 1/a(k).

%C Log(Phi) = (125*C - 55) / (24*sqrt(5)), where C = Sum_{k>=1} (-1)^(k+1)*1/a(k). See A002390 = Decimal expansion of natural logarithm of golden ratio.

%C 3-d analog of the Catalan numbers: (3n)!/(n!(n+1)!(n+2)!) = A161581(n) = A006480(n) / ((n+1)^2*(n+2)), where A006480(n) = (3n)!/(n!)^3 De Bruijn's S(3,n). (End)

%C For a relation to the inviscid Burgers', or Hopf, equation, see A001764. - _Tom Copeland_, Feb 15 2014

%C From _Fung Lam_, May 01 2014: (Start)

%C One class of generalized Catalan numbers can be defined by g.f. A(x) = (1-sqrt(1-q*4*x*(1-(q-1)*x)))/(2*q*x) with nonzero parameter q. Recurrence: (n+3)*a(n+2) -2*q*(2*n+3)*a(n+1) +4*q*(q-1)*n*a(n) = 0 with a(0)=1, a(1)=1.

%C Asymptotic approximation for q >= 1: a(n) ~ (2*q+2*sqrt(q))^n*sqrt(2*q*(1+sqrt(q))) /sqrt(4*q^2*Pi*n^3).

%C For q <= -1, the g.f. defines signed sequences with asymptotic approximation: a(n) ~ Re(sqrt(2*q*(1+sqrt(q)))*(2*q+2*sqrt(q))^n) / sqrt(q^2*Pi*n^3), where Re denotes the real part. Due to Stokes' phenomena, accuracy of the asymptotic approximation deteriorates at/near certain values of n.

%C Special cases are A000108 (q=1), A068764 to A068772 (q=2 to 10), A240880 (q=-3).

%C (End)

%C Number of sequences [s(0), s(1), ..., s(n)] with s(n)=0, Sum_{j=0..n} s(j) = n, and Sum_{j=0..k} s(j)-1 >= 0 for k<n-1 (and necessarily Sum_{j=0..n-1} s(j)-1 = 0). These are the branching sequences of the (ordered) trees with n non-root nodes, see example. - _Joerg Arndt_, Jun 30 2014

%C Number of stack-sortable permutations of [n], these are the 231-avoiding permutations; see the Bousquet-Mélou reference. - _Joerg Arndt_, Jul 01 2014

%C a(n) is the number of increasing strict binary trees with 2n-1 nodes that avoid 132. For more information about increasing strict binary trees with an associated permutation, see A245894. - _Manda Riehl_, Aug 07 2014

%C In a one-dimensional medium with elastic scattering (zig-zag walk), first recurrence after 2n+1 scattering events has the probability C(n)/2^(2n+1). - _Joachim Wuttke_, Sep 11 2014

%C The o.g.f. C(x) = [1 - sqrt(1-4x)]/2, for the Catalan numbers, with comp. inverse Cinv(x) = x*(1-x) and the functions P(x) = x / (1 + t*x) and its inverse Pinv(x,t) = -P(-x,t) = x / (1 - t*x) form a group under composition that generates or interpolates among many classic arrays, such as the Motzkin (Riordan, A005043), Fibonacci (A000045), and Fine (A000957) numbers and polynomials (A030528), and enumerating arrays for Motzkin, Dyck, and Łukasiewicz lattice paths and different types of trees and non-crossing partitions (A091867, connected to sums of the refined Narayana numbers A134264). - _Tom Copeland_, Nov 04 2014

%C Conjecture: All the rational numbers Sum_{i=j..k} 1/a(i) with 0 < min{2,k} <= j <= k have pairwise distinct fractional parts. - _Zhi-Wei Sun_, Sep 24 2015

%C The Catalan number series A000108(n+3), offset n=0, gives Hankel transform revealing the square pyramidal numbers starting at 5, A000330(n+2),offset n=0 (Empirical observation). - _Tony Foster III_, Sep 05 2016

%C Hankel transforms of the Catalan numbers with the first 2, 4, and 5 terms omitted give A001477, A006858, and A091962, respectively, without the first 2 terms in all cases. More generally, the Hankel transform of the Catalan numbers with the first k terms omitted is H_k(n) = Prod_{ j=1..k-1 } Prod_{ i=1..j } (2*n+j+i)/(j+i) [see Cigler (2011), Eq. (1.14) and references therein]. - _Andrey Zabolotskiy_, Oct 13 2016

%C Presumably this satisfies Benford's law, although the results in Hürlimann (2009) do not make this clear. See S. J. Miller, ed., 2015, p. 5. - _N. J. A. Sloane_, Feb 09 2017

%D The large number of references and links demonstrates the ubiquity of the Catalan numbers.

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%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 333 and p. 337.

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%H H. Bottomley, <a href="/A000108/a000108b.gif">Catalan Space Invaders</a>

%H H. Bottomley, <a href="/A002694/a002694.gif">Illustration for A000108, A001147, A002694, A067310 and A067311</a>

%H T. Bourgeron, <a href="http://www.dma.ens.fr/culturemath/maths/pdf/combi/montagnards.pdf">Montagnards et polygones</a>

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%H H Cambazard, N Catusse, <a href="http://arxiv.org/abs/1512.06649">Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the Plane</a>, arXiv preprint arXiv:1512.06649, 2015

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%H Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.

%H P. J. Cameron, <a href="http://dx.doi.org/10.1093/qmath/38.2.155">Some treelike objects</a>, Quart. J. Math. Oxford, 38 (1987), 155-183. See p. 162

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H A. Cayley, <a href="http://dx.doi.org/10.1112/plms/s1-22.1.237">On the partitions of a polygon</a>, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.

%H F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies in Automatic Combinatorics, Volume II (1997).

%H José Luis Cereceda, <a href="http://arxiv.org/abs/1510.00731">An alternative recursive formula for the sums of powers of integers</a>, arXiv:1510.00731 [math.CO], 2015.

%H G. Chatel, V. Pilaud, <a href="http://arxiv.org/abs/1411.3704">The Cambrian and Baxter-Cambrian Hopf Algebras</a>, arXiv preprint arXiv:1411.3704 [math.CO], 2014.

%H Julie Christophe, Jean-Paul Doignon and Samuel Fiorini, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Doignon/doignon40.html">Counting Biorders</a>, J. Integer Seqs., Vol. 6, 2003.

%H J. Cigler, <a href="http://arxiv.org/abs/1109.1449">Some nice Hankel determinants</a>, arXiv:1109.1449 [math.CO], 2011.

%H J. Cigler, <a href="http://homepage.univie.ac.at/Johann.Cigler/preprints/chebyshev-survey.pdf">Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results</a>, 2013.

%H Alissa S. Crans, <a href="https://www.youtube.com/watch?v=eoofvKI_Okg">A surreptitious sequence: the Catalan numbers</a> (2014)

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%H Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, Leon C. Woodson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Davenport/dav3.html">The Boundary of Ordered Trees</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.

%H T. Davis, <a href="http://www.geometer.org/mathcircles/catalan.pdf">Catalan Numbers</a>

%H E. Deutsch and B. E. Sagan, <a href="http://arxiv.org/abs/math.CO/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, J. Num. Theory 117 (2006), 191-215.

%H R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/catalan.html">Catalan numbers</a> (<a href="http://www-groups.dcs.st-andrews.ac.uk/~history/Miscellaneous/CatalanNumbers/catalan.html">another copy</a>)

%H T. Dokos, I. Pak, <a href="http://arxiv.org/abs/1401.0770">The expected shape of random doubly alternating Baxter permutations</a>, arXiv:1401.0770 [math.CO], 2014.

%H T. Doslic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Doslic/doslic6.html">Handshakes across a (round) table</a>, JIS 13 (2010) #10.2.7.

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%H P. Tarau, <a href="http://arxiv.org/abs/1507.06944">A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations</a>, arXiv preprint arXiv:1507.06944 [cs.LO], 2015.

%H D. Taylor, <a href="http://www.maths.usyd.edu.au/u/don/code/Catalan/Catalan.html">Catalan Structures(up to C(7))</a>

%H J.-D. Urbina, J. Kuipers, Q. Hummel, K. Richter, <a href="http://arxiv.org/abs/1409.1558">Multiparticle correlations in complex scattering and the mesoscopic Boson Sampling problem</a>, arXiv preprint arXiv:1409.1558 [quant-ph], 2014.

%H A. Vieru, <a href="http://arxiv.org/abs/1107.2938">Agoh's conjecture: its proof, its generalizations, its analogues</a>, arXiv:1107.2938 [math.NT], 2011.

%H Gérard Villemin, <a href="http://villemin.gerard.free.fr/aNombre/TYPDENOM/Catalan/Catalan.htm">Nombres De Catalan</a> (French)

%H D. W. Walkup, <a href="http://dx.doi.org/10.1112/S0025579300005659">The number of plane trees</a>, Mathematika, vol. 19, No. 2 (1972), 200-204.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>, <a href="http://mathworld.wolfram.com/BinaryBracketing.html">Binary Bracketing</a>, <a href="http://mathworld.wolfram.com/BinaryTree.html">Binary Tree</a>, <a href="http://mathworld.wolfram.com/NonassociativeProduct.html">Nonassociative Product</a>, <a href="http://mathworld.wolfram.com/StaircaseWalk.html">Staircase Walk</a>, and <a href="http://mathworld.wolfram.com/DyckPath.html">Dyck Path</a>

%H J. Winter, M. M. Bonsangue and J. J. M. M. Rutten, <a href="http://oai.cwi.nl/oai/asset/21313/21313A.pdf ">Context-free coalgebras</a>, 2013

%H Roman Witula, Damian Slota and Edyta Hetmaniok, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from255to263.pdf">Bridges between different known integer sequences</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 255-263.

%H W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WOAN/hankel2.html">Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001), #01.1.2.

%H Wen-jin Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Woan/woan11.html">A Recursive Relation for Weighted Motzkin Sequences</a> Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.6.

%H Wen-jin Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Woan2/woan35.html">Animals and 2-Motzkin Paths</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.6.

%H Wen-jin Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Woan/woan206.html">A Relation Between Restricted and Unrestricted Weighted Motzkin Paths</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.7.

%H F. Yano and H. Yoshida, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.050">Some set partition statistics in non-crossing partitions and generating functions</a>, Discr. Math., 307 (2007), 3147-3160.

%H Yan X Zhang, <a href="http://arxiv.org/abs/1508.00318">Four Variations on Graded Posets</a>, arXiv preprint arXiv:1508.00318 [math.CO], 2015.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>

%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>

%F a(n) = A000984(n)/(n+1) = binomial(2*n, n)/(n+1) = (2*n)!/(n!*(n+1)!).

%F a(n) = binomial(2*n, n)-binomial(2*n, n-1).

%F a(n) = Sum_{k=0..n-1} a(k)a(n-1-k).

%F a(n) = Product_{k=2..n} (1 + n/k), if n>1.

%F G.f.: A(x) = (1 - sqrt(1 - 4*x)) / (2*x). G.f. A(x) satisfies A = 1 + x*A^2.

%F a(n+1) = Sum_{i} binomial(n, 2*i)*2^(n-2*i)*a(i). - Touchard

%F 2*(2*n-1)*a(n-1) = (n+1)*a(n).

%F It is known that a(n) is odd if and only if n=2^k-1, k=0, 1, 2, 3, ... - _Emeric Deutsch_, Aug 04 2002, corrected by _M. F. Hasler_, Nov 08 2015

%F 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

%F Integral representation: a(n) = int(x^n*sqrt((4-x)/x), x=0..4)/(2*Pi). - _Karol A. Penson_, Apr 12 2001

%F E.g.f.: exp(2*x)*(I_0(2*x)-I_1(2*x)), where I_n is Bessel function. - _Karol A. Penson_, Oct 07 2001

%F Polygorial(n, 6)/Polygorial(n, 3). - Daniel Dockery (peritus(AT)gmail.com), Jun 24 2003

%F G.f. A(x) satisfies ((A(x) + A(-x)) / 2)^2 = A(4*x^2). - _Michael Somos_, Jun 27, 2003

%F G.f. A(x) satisfies Sum_{k>=1} k(A(x)-1)^k = Sum_{n >= 1} 4^{n-1}*x^n. - Shapiro, Woan, Getu

%F a(n+m) = Sum_{k} A039599(n, k)*A039599(m, k). - _Philippe Deléham_, Dec 22 2003

%F a(n+1) = (1/(n+1))*Sum_{k=0..n} a(n-k)*binomial(2k+1, k+1). - _Philippe Deléham_, Jan 24 2004

%F a(n) = Sum_{k>=0} A008313(n, k)^2. - _Philippe Deléham_, Feb 14 2004

%F a(m+n+1) = Sum_{k>=0} A039598(m, k)*A039598(n, k). - _Philippe Deléham_, Feb 15 2004

%F C(n-1) = binomial(2*n-2,n-1)/n = (1/n!) * [ n^(n-1) + ( binomial(n-2,1) + binomial(n-2,2) )*n^(n-2) + ( 2*binomial(n-3,1) + 7*binomial(n-3,2) + 8*binomial(n-3,3) + 3*binomial(n-3,4) )*n^(n-3) + ( 6*binomial(n-4,1) + 38*binomial(n-4,2) + 93*binomial(n-4,3) + 111*binomial(n-4,4) + 65*binomial(n-4,5) + 15*binomial(n-4,6) )*n^(n-4) + ... ]. - _André F. Labossière_, Nov 10 2004, corrected by _M. F. Hasler_, Nov 10 2015

%F a(n) = Sum_{k=0..n} (-1)^k*2^(n-k)*binomial(n, k)*binomial(k, floor(k/2)). - Paul Barry, Jan 27 2005

%F Sum_{n>=0} 1/a(n) = 2 + 4*Pi/3^(5/2) = F(1,2;1/2;1/4) = 2.806133050770763... (see L'Univers de Pi link). - _Gerald McGarvey_ and _Benoit Cloitre_, Feb 13 2005

%F a(n) = Sum_{k=0..floor(n/2)} ((n-2*k+1)*binomial(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

%F a((m+n)/2) = Sum_{k>=0} A053121(m, k)*A053121(n, k) if m+n is even. - _Philippe Deléham_, May 26 2005

%F E.g.f. Sum_{n>=0} a(n) * x^(2*n) / (2*n)! = BesselI(1, 2*x) / x. - _Michael Somos_, Jun 22 2005

%F Given g.f. A(x), then B(x) = x * A(x^3) satisfies 0 = f(x, B(X)) where f(u, v) = u - v + (u*v)^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

%F 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)). - _Franklin T. Adams-Watters_, Feb 08 2006

%F Sum_{k>=1} a(k)/4^k = 1. - _Franklin T. Adams-Watters_, Jun 28 2006

%F a(n) = A047996(2*n+1, n). - _Philippe Deléham_, Jul 25 2006

%F Binomial transform of A005043. - _Philippe Deléham_, Oct 20 2006

%F a(n) = Sum_{k=0..n} (-1)^k*A116395(n,k). - _Philippe Deléham_, Nov 07 2006

%F 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].

%F a(k) = Sum_{i=1..k} |A008276(i,k)| * (k-1)^(k-i) / k!. - _André F. Labossière_, May 29 2007

%F a(n) = Sum_{k=0..n} A129818(n,k) * A007852(k+1). - _Philippe Deléham_, Jun 20 2007

%F a(n) = Sum_{k=0..n} A109466(n,k) * A127632(k). - _Philippe Deléham_, Jun 20 2007

%F Row sums of triangle A124926. - _Gary W. Adamson_, Oct 22 2007

%F Lim_{n->infinity}(1+Sum_{k=0..n}a(k)/A004171(k)) = 4/Pi. - _Reinhard Zumkeller_, Aug 26 2008

%F a(n) = Sum_{k=0..n} A120730(n,k)^2 and a(k+1) = Sum_{n>=k} A120730(n,k). - _Philippe Deléham_, Oct 18 2008

%F 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])). - _Gary W. Adamson_, Oct 27 2008

%F 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. - _Thomas Wieder_, Feb 25 2009

%F Let A(x) be the g.f., then B(x)=x*A(x) satisfies the differential equation B'(x)-2*B'(x)*B(x)-1=0. - _Vladimir Kruchinin_, Jan 18 2011

%F G.f.: 1/(1-x/(1-x/(1-x/(...)))) (continued fraction). - _Joerg Arndt_, Mar 18 2011

%F 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). - _Tom Copeland_, Sep 04 2011

%F 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. - _Tom Copeland_, Sep 04 2011

%F 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 and this relates the Catalan numbers to the row sums of A125181. - _Tom Copeland_, Sep 30 2011

%F 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)). - _Tom Copeland_, Sep 30 2011

%F G.f.: (1-sqrt(1-4*x))/(2*x)=G(0) where 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

%F E.g.f.: exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x))=G(0) where 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

%F 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

%F a(n) = A208355(2*n-1) = A208355(2*n) for n > 0. - _Reinhard Zumkeller_, Mar 04 2012

%F G.f.: 1 + 2*x/(U(0)-2*x) where U(k)= k*(4*x+1) + 2*x + 2 - x*(2*k+3)*(2*k+4)/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Sep 20 2012

%F G.f.: hypergeom([1/2,1],[2],4*x). - _Joerg Arndt_, Apr 06 2013

%F Special values of Jacobi polynomials, in Maple notation: a(n) = 4^n*JacobiP(n,1,-1/2-n,-1)/(n+1). - _Karol A. Penson_, Jul 28 2013

%F For n > 0: a(n) = sum of row n in triangle A001263. - _Reinhard Zumkeller_, Oct 10 2013

%F a(n) = binomial(2n,n-1)/n and a(n) mod n = binomial(2n,n) mod n = A059288(n). - _Jonathan Sondow_, Dec 14 2013

%F a(n-1) = sum(t1+2*t2+...+n*tn=n, (-1)^(1+t1+t2+...+tn)*multinomial(t1+t2 +...+tn,t1,t2,...,tn)*a(1)^t1*a(2)^t2*...*a(n)^tn). - _Mircea Merca_, Feb 27 2014

%F a(n) = sum_{k=1..n} binomial(n+k-1,n)/n if n>0. _Alexander Adamchuk_, Mar 25 2014

%F a(n) = -2^(2*n+1) * binomial(n-1/2, -3/2). - _Peter Luschny_, May 06 2014

%F a(n) = (4*A000984(n)-A000984(n+1))/2. - _Stanislav Sykora_, Aug 09 2014

%F a(n) = A246458(n) * A246466(n). - _Tom Edgar_, Sep 02 2014

%F a(n) = (2*n)!*[x^(2*n)]hypergeom([],[2],x^2). - _Peter Luschny_, Jan 31 2015

%F a(n) = 4^(n-1)*hypergeom([3/2, 1-n], [3], 1). - _Peter Luschny_, Feb 03 2015

%F a(2n) = 2*A000150(2n); a(2n+1) = 2*A000150(2n+1) + a(n). - _John Bodeen_, Jun 24 2015

%F a(n) = Sum_{t=1, n+1} n^(t-1)*abs(stirling1(n+1, t)) / Sum_{t=1, n+1} abs(stirling1(n+1, t)), for n >0, see (10) in Cereceda link. - _Michel Marcus_, Oct 06 2015

%F a(n) ~ 4^(n-2)*(128+160/N^2+84/N^4+715/N^6-10180/N^8)/(N^(3/2)*Pi^(1/2)) where N=4*n+3. - _Peter Luschny_, Oct 14 2015

%F a(n) = sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*binomial(n+1-k,k)*a(n-k) if n > 0; and a(0) = 1. - _David Pasino_, Jun 29 2016

%F Sum_{n>=0} (-1)^n/a(n) = 14/25 - 24*arccsch(2)/(25*sqrt(5)) = 14/25 - 24*A002390/(25*sqrt(5)) = 0.353403708337278061333... - _Ilya Gutkovskiy_, Jun 30 2016

%F C(n) = (1/n) * Sum_{i+j+k=n-1} C(i)*C(j)*C(k)*(k+1), n>=1. - _Yuchun Ji_, Feb 21 2016

%F C(n) = 1 + Sum(C(i)*C(j)*C(k), i+j+k<n-1) - Yuchun Ji, Sep 1 2016

%e G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ...

%e From _Joerg Arndt_ and Greg Stevenson, Jul 11 2011: (Start)

%e The following products of 3 transpositions lead to a 4-cycle in S_4:

%e (1,2)*(1,3)*(1,4);

%e (1,2)*(1,4)*(3,4);

%e (1,3)*(1,4)*(2,3);

%e (1,4)*(2,3)*(2,4);

%e (1,4)*(2,4)*(3,4). (End)

%e For n=3, a(3)=5 since there are exactly 5 binary sequences of length 7 in which the number of ones first exceed the number of zeros at entry 7, namely, 0001111, 0010111, 0011011, 0100111, and 0101011. - _Dennis P. Walsh_, Apr 11 2012

%e From _Joerg Arndt_, Jun 30 2014: (Start)

%e The a(4) = 14 branching sequences of the (ordered) trees with 4 non-root nodes are (dots denote zeros):

%e 01: [ 1 1 1 1 . ]

%e 02: [ 1 1 2 . . ]

%e 03: [ 1 2 . 1 . ]

%e 04: [ 1 2 1 . . ]

%e 05: [ 1 3 . . . ]

%e 06: [ 2 . 1 1 . ]

%e 07: [ 2 . 2 . . ]

%e 08: [ 2 1 . 1 . ]

%e 09: [ 2 1 1 . . ]

%e 10: [ 2 2 . . . ]

%e 11: [ 3 . . 1 . ]

%e 12: [ 3 . 1 . . ]

%e 13: [ 3 1 . . . ]

%e 14: [ 4 . . . . ]

%e (End)

%p A000108 := n->binomial(2*n,n)/(n+1); G000108 := (1 - sqrt(1 - 4*x)) / (2*x);

%p spec := [ A, {A=Prod(Z,Sequence(A))}, unlabeled ]: [ seq(combstruct[count](spec, size=n+1), n=0..42) ];

%p with(combstruct):bin := {B=Union(Z,Prod(B,B))}: seq(count([B,bin,unlabeled],size=n), n=1..25); # _Zerinvary Lajos_, Dec 05 2007

%p 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_, May 21 2008

%p seq((2*n)!*coeff(series(hypergeom([],[2],x^2),x,2*n+2),x,2*n),n=0..30); # _Peter Luschny_, Jan 31 2015

%t (* TermFunction *)

%t CatalanNumber

%t (* TermFunctionDefinition *)

%t A000108[n_] := (2 n)!/n!/(n+1)!

%t (* TermFunctionDefinition *)

%t A000108[n_] := Hypergeometric2F1[1 - n, -n, 2, 1] (* Richard L. Ollerton, Sep 13 2006 *)

%t (* TermList *)

%t Table[ CatalanNumber@ n, {n, 0, 24}] (* _Robert G. Wilson v_, Feb 15 2011 *)

%t (* TermList *)

%t CoefficientList[InverseSeries[Series[x/Sum[x^n, {n, 0, 31}], {x, 0, 31}]]/x, x] (* _Mats Granvik_, Nov 24 2013 *)

%t (* TermListByIndexFunction *)

%t Function[n, CatalanNumber /@ Range[0, n]]

%o (PARI) a(n)=binomial(2*n,n)/(n+1) \\ _M. F. Hasler_, Aug 25 2012

%o (PARI) a(n) = (2*n)! / n! / (n+1)!

%o (PARI) a(n) = my(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));

%o (PARI) {a(n) = if( n<1, n==0, polcoeff( serreverse( x / (1 + x)^2 + x * O(x^n)), n))}; /* _Michael Somos_ */

%o (PARI) (recur(a,b)=if(b<=2,(a==2)+(a==b)+(a!=b)*(1+a/2), (1+a/b)*recur(a,b-1))); a(n)=recur(n,n); \\ _R. J. Cano_, Nov 22 2012

%o (PARI) x='x+O('x^40); Vec((1-sqrt(1-4*x))/(2*x)) \\ _Altug Alkan_, Oct 13 2015

%o (Mupad) combinat::dyckWords::count(n) $ n = 0..38 // _Zerinvary Lajos_, Apr 14 2007

%o (MAGMA) C:= func< n | Binomial(2*n,n)/(n+1) >; [ C(n) : n in [0..60]];

%o (MAGMA) [Catalan(n): n in [0..40]]; // _Vincenzo Librandi_, Apr 02 2011

%o (Haskell)

%o import Data.List (genericIndex)

%o a000108 n = genericIndex a000108_list n

%o a000108_list = 1 : catalan [1] where

%o catalan cs = c : catalan (c:cs) where

%o c = sum $ zipWith (*) cs $ reverse cs

%o -- _Reinhard Zumkeller_, Nov 12 2011

%o a000108 = map last $ iterate (scanl1 (+) . (++ [0])) [1]

%o -- _David Spies_, Aug 23 2015

%o (Sage) [catalan_number(i) for i in range(27)] # _Zerinvary Lajos_, Jun 26 2008

%o (Sage) [binomial(2*i,i)-binomial(2*i,i-1) for i in xrange(0,25)] # _Zerinvary Lajos_, May 17 2009

%o (Sage) # Generalized algorithm of L. Seidel

%o def A000108_list(n) :

%o D = [0]*(n+1); D[1] = 1

%o b = True; h = 1; R = []

%o for i in range(2*n-1) :

%o if b :

%o for k in range(h,0,-1) : D[k] += D[k-1]

%o h += 1; R.append(D[1])

%o else :

%o for k in range(1,h, 1) : D[k] += D[k+1]

%o b = not b

%o return R

%o A000108_list(31) # _Peter Luschny_, Jun 02 2012

%o (Maxima) A000108(n):=binomial(2*n,n)/(n+1)$ makelist(A000108(n),n,0,30); /* _Martin Ettl_, Oct 24 2012 */

%o (Python)

%o from gmpy2 import divexact

%o A000108 = [1, 1]

%o for n in range(1,10**3):

%o ....A000108.append(divexact(A000108[-1]*(4*n+2),(n+2))) # _Chai Wah Wu_, Aug 31 2014

%Y Cf. A000142, A000245, A000344, A000588, A000957, A000984, A001392, A001453, A001791, A002057, A002420, A003046, A003517, A003518, A003519, A006480, A008276, A008549, A014137, A014138, A014140, A022553, A024492, A032357, A032443, A039599, A048990, A059288, A068875, A069640, A086117, A094216, A094638, A094639, A098597, A099731, A119822, A120304, A124926, A129763, A137697, A154559, A161581, A167892, A167893, A179277, A211611.

%Y A row of A060854.

%Y See A001003, A001190, A001699, A000081 for other ways to count parentheses.

%Y Enumerates objects encoded by A014486.

%Y A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

%Y Cf. A051168 (diagonal of the square array described).

%Y Cf. A033552, A176137 (partitions into Catalan numbers).

%Y Cf. A000753, A000736 (Boustrophedon transforms).

%Y Cf. A120303 (largest prime factor of Catalan number).

%Y Cf. A121839 (reciprocal Catalan constant).

%Y Cf. A038003, A119861, A119908, A120274, A120275 (odd Catalan number).

%Y Cf. A002390 (decimal expansion of natural logarithm of golden ratio).

%Y Coefficients of square root of the g.f. are A001795/A046161.

%Y For a(n) mod 6 see A259667.

%Y For a(n) in base 2 see A264663.

%K core,nonn,easy,eigen,nice

%O 0,3

%A _N. J. A. Sloane_

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Last modified May 23 18:42 EDT 2017. Contains 286926 sequences.