0,2

Equal to the binomial coefficient sum Sum_{k=0..n} binomial(n,k)^2.

Number of possible interleavings of a program with n atomic instructions when executed by two processes - Manuel Carro (mcarro(AT)fi.upm.es), Sep 22 2001

Convolving a(n) with itself yields A000302, the powers of 4. - T. D. Noe (noe(AT)sspectra.com), Jun 11 2002

a(n)=Max{ (i+j)!/(i!j!) | 0<=i,j<=n } - Benoit Cloitre (benoit7848c(AT)orange.fr), May 30 2002

Number of ordered trees with 2n+1 edges, having root of odd degree and nonroot nodes of outdegree 0 or 2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 02 2002

Also number of directed, convex polyominoes having semiperimeter n+2.

Also number of diagonally symmetric, directed, convex polyominoes having semiperimeter 2n+2. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 03 2002

Also Sum_{k=0..n} binomial(n+k-1,k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 28 2002

The second inverse binomial transform of this sequence is this sequence with interpolated zeros. Its G.f. is (1 - 4*x^2)^(-1/2), with n-th term C(n,n/2)(1+(-1)^n)/2. - Paul Barry (pbarry(AT)wit.ie), Jul 01 2003

Number of possible values of a 2*n bit binary number for which half the bits are on and half are off. - Gavin Scott (gavin(AT)allegro.com), Aug 09 2003

Ordered partitions of n with zeros to n+1, e.g. for n=4 we consider the ordered partitions of 11110 (5), 11200 (30), 13000 (20), 40000 (5) and 22000 (10), total 70 and a(4)=70. See A001700 (esp. Mambetov Bektur's comment). - Jon Perry (perry(AT)globalnet.co.uk), Aug 10 2003

Number of non-decreasing sequences of n integers from 0 to n: a(n) = sum_{i_{1}=0}^{n} sum_{i_{2}=i_{1}}^{n}...sum_{i_{n}=i_{n-1}}^{n}(1). - J. N. Bearden (jnb(AT)eller.arizona.edu), Sep 16 2003

Number of peaks at odd level in all Dyck paths of semilength n+1. Example: a(2)=6 because we have U*DU*DU*D, U*DUUDD, UUDDU*D, UUDUDD, UUU*DDD, where U=(1,1), D=(1,-1) and * indicates a peak at odd level. Number of ascents of length 1 in all Dyck paths of semilength n+1 (an ascent in a Dyck path is a maximal string of up steps). Example: a(2)=6 because we have uDuDuD, uDUUDD, UUDDuD, UUDuDD, UUUDDD, where an ascent of length 1 is indicated by a lower case letter. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 05 2003

a(n-1)=number of subsets of 2n-1 distinct elements taken n at a time that contain a given element. e.g. n=4 -> a(3)=20 and if we consider the subsets of 7 taken 4 at a time with a 1 we get (1234, 1235, 1236, 1237, 1245, 1246, 1247, 1256, 1257, 1267, 1345, 1346, 1347, 1356, 1357, 1367, 1456, 1457, 1467, 1567) and there are 20 of them. - Jon Perry (perry(AT)globalnet.co.uk), Jan 20 2004

The dimension of a particular (necessarily existent) absolutely universal embedding of the unitary dual polar space DSU(2n,q^2) where q>2. - J. Taylor (jt_cpp(AT)yahoo.com), Apr 02 2004.

Number of standard tableaux of shape (n+1, 1^n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004

Erdos, Graham et al. conjectured that a(n) is never squarefree for sufficiently large n. Sarkozy showed that if s(n) is the square part of a(n), then s(n) is asymptotically (sqrt(2)-2)*(sqrt(n))*(Riemann Zeta Function(1/2)). Granville and Ramare proved that the only squarefree values are a(1)=2, a(2)=6 and a(4)=70. A000984(n)/(n+1) = A000108(n), that is, dividing by (n+1) scales the Central binomial coefficients to Catalan numbers also called Segner numbers. - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 04 2004

p divides a((p-1)/2)-1=A030662[n] for prime p=5,13,17,29,37,41,53,61,73,89,97..=A002144[n] Pythagorean primes: primes of form 4n+1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 04 2006

The number of direct routes from my home to Granny's when Granny lives n blocks south and n blocks east of my home in Grid City. To obtain a direct route, from the 2n blocks, choose n blocks on which one travels south. For example, a(2)=6 because there are 6 direct routes: SSEE, SESE, SEES, EESS, ESES and ESSE. - Dennis P. Walsh (dwalsh(AT)mtsu.edu), Oct 27 2006

Inverse: With q = -log(log(16)/(pi a(n)^2)), ceiling((q + log(q))/log(16)) = n. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 26 2007

Number of partitions with Ferrers diagrams that fit in an n X n box (including the empty partition of 0). Example: a(2) = 6 because we have: empty, 1, 2, 11, 21 and 22. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 02 2007

The number of walks of length 2n on an infinite linear lattice that begin and end at the origin. - Stefan Hollos (stefan(AT)exstrom.com), Dec 10 2007

Integral representation : C(2n,n)=1/Pi Integral [(2x)^(2n)/Sqrt[1 - x^2],{x,-1, 1}], i.e. C(2n,n)/4^n is the moment of order 2n of the arcsin distribution on the interval (-1,1). - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Jan 02 2008

Define the array m(1,j)=1 ; m(i,1)=1 ; m(i,j)=m(i,j-1) + m(j,i-1), then a(n) = m(n,n) [From philippe lallouet (philip.lallouet(AT)orange.fr), Sep 15 2008]

Also the Catalan transform of A000079. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 06 2008]

Straub, Amdeberhan and Moll: "... it is conjectured that there are only finitely many indices n such that C_n is not divisible by any of 3, 5, 7 and 11. Finally, we remark that Erdos et al. conjectured that the central binomial coefficients C_n are never squarefree for n > 4 which has been proved by Granville and Ramare." [From Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 14 2008]

Equals row sums of triangle A152229 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]

Equals row sums of triangle A158815 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 27 2009]

This sequence appears in formulae in the link cited. [Oktay Haracci (timetunnel3(AT)hotmail.com), Apr 02 2009]

Equals INVERT transform of A081696: (1, 1, 3, 9, 29, 97, 333,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

Also, in sports, the number of ordered ways for a "Best of 2n-1 Series" to progress. For example, a(2) = 6 means there are six ordered ways for a "best of 3" series to progress. If we write A for a win by "team A" and B for a win by "team B" and if we list the played games chronologically from left to right then the six ways are AA, ABA, BAA, BB, BAB, and ABB. (Proof: To generate the a(n) ordered ways: Write down all a(n) ways to designate n of 2n games as won by team A. Remove the maximal suffix of identical letters from each of these.) [From Lee A. Newberg (integer(AT)quantconsulting.com), Jun 02 2009]

Contribution from Jason Richardson-White (coyoteworks(AT)gmail.com), Jun 15 2009: Index the central binomial coefficients with the natural numbers 1,2,3...,n. It appears that dividing the central binomial coefficients by their indexes yields the Catalan numbers (A000108).

Number of nXn binary arrays with rows, considered as binary numbers, in nondecreasing order, and columns, considered as binary numbers, in nonincreasing order. [From Ron Hardin (rhhardin(AT)att.net), Jun 27 2009]

Hankel transform is 2^n. [From Paul Barry (pbarry(AT)wit.ie), Aug 05 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 160.

A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, preprint, 2007.

Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.

Thierry Dana-Picard, Sequences of Definite Integrals, Factorials and Double Factorials, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.6.

E. Deutsch and L. Shapiro, Seventeen Catalan identities, Bulletin of the Institute of Combinatorics and its Applications, 31, 31-38, 2001.

Erdos, P.; Graham, R. L.; Ruzsa, I. Z.; and Straus, E. G. "On the Prime Factors of C(2n,n)." Math. Comput. 29, 83-92, 1975.

H. W. Gould, Combinatorial Identities, Morgantown, 1972, (3.66), page 30.

M. Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics Trust (2008), 3-124. [From Martin Griffiths (griffm(AT)essex.ac.uk), Mar 28 2009]

Granville, A. and Ramare, O. "Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43, 73-107, 1996.

J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Sarkozy, A. "On Divisors of Binomial Coefficients. I." J. Number Th. 20, 70-80, 1985.

L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

T. D. Noe, Table of n, a(n) for n = 0..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].

D. H. Bailey, J. M. Borwein and D. M. Bradley, Experimental determination of Ap'ery-like identities for zeta(4n+2)

J. Borwein and D. Bradley, Empirically determined Ap'ery-like formulae for zeta(4n+3)

N. T. Cameron, Random walks, trees and extensions of Riordan group techniques

B. N. Cooperstein and E. E. Shult, A note on embedding and generating dual polar spaces. Adv. Geom. 1 (2001), 37-48. See Theorem 5.4.

R. M. Dickau, Shortest-path diagrams

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 77

Oktay Haracci (timetunnel3(AT)hotmail.com), Regular Polygons

Ron Hardin, Binary arrays with both rows and cols sorted, symmetries

Milan Janjic, Two Enumerative Functions

I. Jensen, Series exapansions for self-avoiding polygons

C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic

P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Armin Straub, Tewodros Amdeberhan and Victor H. Moll, The p-adic valuation of k-central binomial coefficients, Nov 13, 2008, pp. 10-11. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 14 2008]

V. Strehl, Recurrences and Legendre transform

R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.

H. A. Verrill, Sums of squares of binomial coefficients, ...

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Circle Line Picking

Index entries for "core" sequences

G.f.: A(x) = (1 - 4*x)^(-1/2) = 1 + 2*x + 6*x^2 + 20*x^3 + ...

a(n) = 2^n/n! * product[ k=0..n-1 ] (2*k+1).

a(n) = a(n-1)*(4-2/n) = 4a(n-1)+A002420(n) = A000142(2n)/(A000142(n)^2) = A001813(n)/A000142(n) = sqrt(A002894(n)) = A010050(n)/A001044(n) = (n+1)*A000108(n) = -A005408(n-1)*A002420(n) - Henry Bottomley (se16(AT)btinternet.com), Nov 10 2000

Using Stirling's formula in A000142 it is easy to get the asymptotic expression a(n) ~ 4^n / sqrt(Pi * n) - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

Integral representation as n-th moment of a positive function on the interval[0, 4], in Maple notation: a(n)= int(x^n*((x*(4-x))^(-1/2))/Pi, x=0..4), n=0, 1, ... This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 17 2001

sum(n>=1, 1/a(n))=(2*Pi*sqrt(3)+9)/27 - Benoit Cloitre (benoit7848c(AT)orange.fr), May 01 2002

E.g.f.: exp(2x) I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 08 2002

E.g.f.: I_0(2x)=sum a(n) x^(2n)/(2n)!, where I_0 is Bessel function. - Michael Somos, Sep 09, 2002.

a(n) = sum(k=0, n, C(n, k)^2). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 31 2003

Determinant of n X n matrix M(i, j)=binomial(n+i, j) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 28 2003

Given m = C(2n, n), let f be the inverse function, so that f(m) = n. Letting q denote -Log(Log(16)/(m^2*Pi)), we have f(m) = Ceiling( (q + Log(q)) / Log(16) ). - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Oct 30 2003

a(n) = 2*Sum{k= 0...(n-1), a(k)*a(n-k+1)/(k+1)}. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 01 2004

a(n+1)=sum(j=n, n*2+1, binomial(j, n)). E.g. a(4)=C(7, 3)+C(6, 3)+C(5, 3)+C(4, 3)+C(3, 3)=35+20+10+4+1=70 - Jon Perry (perry(AT)globalnet.co.uk), Jan 20 2004

a(n) = (-1)^(n)*sum(j=0..(2*n), (-1)^j*binomial(2*n, j)^2) - Helena Verrill (verrill(AT)math.lsu.edu), Jul 12 2004

a(n)=sum{k=0..n, binomial(2n+1, k)*sin((2n-2k+1)*pi/2)}. - Paul Barry (pbarry(AT)wit.ie), Nov 02 2004

a(n-1)=(1/2)*(-1)^n*sum_{0<=i, j<=n}(-1)^(i+j)*binomial(2n, i+j) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 18 2005

a(n) = C(2n, n-1) + C(n) = A001791(n) + A000108(n). a(n) = (n+1)*C(n) = (n+1)*A000108(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 02 2005

G.f.: c(x)^2/(2*c(x)-c(x)^2) where c(x) is the g.f. of A000108; - Paul Barry (pbarry(AT)wit.ie), Feb 03 2006

a(n)=A006480(n)/A005809(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007

a(n)=Sum{k, 0<=k<=n}A106566(n,k)*2^k. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 25 2007

a(n)= Sum{k>=0, A039599(n, k)} . a(n)= Sum{k>=0, A050165(n, k)} . a(n)= Sum{k>=0, A059365(n, k)*2^k}, n>0 . a(n+1)= Sum{k>=0, A009766(n, k)*2^(n-k+1)}. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 01 2004

a(n)=4^n*sum{k=0..n, C(n,k)(-4)^(-k)*A000108(n+k)}; - Paul Barry (pbarry(AT)wit.ie), Oct 18 2007

Row sums of triangle A135091 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 18 2007

a(n)=Sum_{k, 0<=k<=n}A039598(n,k)*A059841(k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 12 2008]

A007814(a(n))=A000120(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jul 20 2009]

Contribution from Paul Barry (pbarry(AT)wit.ie), Aug 05 2009: (Start)

G.f.: 1/(1-2x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction);

G.f.: 1/(1-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)

a(n)=Product(k=1..n)[4-2/k] [From David Brown (thedabs(AT)gmail.com), Sep 19 2009]

A000984 := n-> binomial(2*n, n);

with(combstruct); [seq(count([S, {S=Prod(Set(Z, card=i), Set(Z, card=i))}, labeled], size=(2*i)), i =0..20)];

with(combstruct); [seq(count([S, {S=Sequence(Union(Arch, Arch)), Arch=Prod(Epsilon, Sequence(Arch), Z)}, unlabeled], size=i), i=0..25)];

Z:=(1-sqrt(1-z))*4^n/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2007

with(combstruct):bin := {B=Union(Z, Prod(B, B))}: seq (count([B, bin, unlabeled], size=n)*n, n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 05 2007

Table[Binomial[2n, n], {n, 0, 24}] (Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 10 2005)

(MAGMA) a:= func< n | Binomial(2*n, n) >; [ a(n) : n in [0..10]];

(PARI) a(n)=if(n<0, 0, (2*n)!/n!^2)

A000984(n+1)=2*A001700(n)=A030662(n)+1. a(2*n) = A001448(n), a(2*n+1) = 2*A002458(n).

Cf. A000108, A002420, A002457. Differs from A071976 at 10-th term.

Bisection of A001405. Row sums of A059481.

Row sums of triangle A008459.

Cf. A030662, A002144.

Cf. A135091.

A152229 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]

A158815 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 27 2009]

A081696 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]

Sequence in context: A056616 A065346 A071976 this_sequence A087433 A119373 A151284

Adjacent sequences: A000981 A000982 A000983 this_sequence A000985 A000986 A000987

nonn,easy,core,nice

N. J. A. Sloane (njas(AT)research.att.com).