%I M4198 N1752 #355 Feb 16 2025 08:32:25

%S 1,6,30,140,630,2772,12012,51480,218790,923780,3879876,16224936,

%T 67603900,280816200,1163381400,4808643120,19835652870,81676217700,

%U 335780006100,1378465288200,5651707681620,23145088600920,94684453367400,386971244197200,1580132580471900

%N a(n) = (2n+1)!/n!^2.

%C Expected number of matches remaining in Banach's modified matchbox problem (counted when last match is drawn from one of the two boxes), multiplied by 4^(n-1). - _Michael Steyer_, Apr 13 2001

%C Hankel transform is (-1)^n*A014480(n). - _Paul Barry_, Apr 26 2009

%C Convolved with A000108: (1, 1, 1, 5, 14, 42, ...) = A000531: (1, 7, 38, 187, 874, ...). - _Gary W. Adamson_, May 14 2009

%C Convolution of A000302 and A000984. - _Philippe Deléham_, May 18 2009

%C 1/a(n) is the integral of (x(1-x))^n on interval [0,1]. Apparently John Wallis computed these integrals for n=0,1,2,3,.... A004731, shifted left by one, gives numerators/denominators of related integrals (1-x^2)^n on interval [0,1]. - _Marc van Leeuwen_, Apr 14 2010

%C Extend the triangular peaks of Dyck paths of semilength n down to the baseline forming (possibly) larger and overlapping triangles. a(n) = sum of areas of these triangles. Also a(n) = triangular(n) * Catalan(n). - _David Scambler_, Nov 25 2010

%C Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n of B equals a(n-1). - _T. D. Noe_, May 01 2011

%C Apparently the number of peaks in all symmetric Dyck paths with semilength 2n+1. - _David Scambler_, Apr 29 2013

%C Denominator of central elements of Leibniz's Harmonic Triangle A003506.

%C Central terms of triangle A116666. - _Reinhard Zumkeller_, Nov 02 2013

%C Number of distinct strings of length 2n+1 using n letters A, n letters B, and 1 letter C. - _Hans Havermann_, May 06 2014

%C Number of edges in the Hasse diagram of the poset of partitions in the n X n box ordered by containment (from Havermann's comment above, C represents the square added in the edge). - _William J. Keith_, Aug 18 2015

%C Let V(n, r) denote the volume of an n-dimensional sphere with radius r then V(n, 1/2^n) = V(n-1, 1/2^n) / a((n-1)/2) for all odd n. - _Peter Luschny_, Oct 12 2015

%C a(n) is the result of processing the n+1 row of Pascal's triangle A007318 with the method of A067056. Example: Let n=3. Given the 4th row of Pascal's triangle 1,4,6,4,1, we get 1*(4+6+4+1) + (1+4)*(6+4+1) + (1+4+6)*(4+1) + (1+4+6+4)*1 = 15+55+55+15 = 140 = a(3). - _J. M. Bergot_, May 26 2017

%C a(n) is the number of (n+1) X 2 Young tableaux with a two horizontal walls between the first and second column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021] and A000984 for one horizontal wall. - _Michael Wallner_, Jan 31 2022

%C a(n) is the number of facets of the symmetric edge polytope of the cycle graph on 2n+1 vertices. - _Mariel Supina_, May 12 2022

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 159.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25; p. 168, #30.

%D W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.

%D C. Jordan, Calculus of Finite Differences. Röttig and Romwalter, Budapest, 1939; Chelsea, NY, 1965, p. 449.

%D M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 127-129.

%D C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514.

%D A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.

%D J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92.

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

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

%D J. Wallis, Operum Mathematicorum, pars altera, Oxford, 1656, pp 31,34 [_Marc van Leeuwen_, Apr 14 2010]

%H G. C. Greubel, <a href="/A002457/b002457.txt">Table of n, a(n) for n = 0..1000</a> [Terms 0 to 200 computed by T. D. Noe; terms 201 to 1000 by G. C. Greubel, Jan 14 2017]

%H Cyril Banderier and Michael Wallner, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/47.html">Young Tableaux with Periodic Walls: Counting with the Density Method</a>, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.

%H Alexander Barg, <a href="https://arxiv.org/abs/2005.12995">Stolarsky's invariance principle for finite metric spaces</a>, arXiv:2005.12995 [math.CO], 2020.

%H W. G. Bickley and J. C. P. Miller, <a href="/A002551/a002551.pdf">Numerical differentiation near the limits of a difference table</a>, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]

%H Sara C. Billey, Matjaž Konvalinka, and Joshua P. Swanson, <a href="https://arxiv.org/abs/1905.00975">Asymptotic normality of the major index on standard tableaux</a>, arXiv:1905.00975 [math.CO], 2019.See p. 15, Remark 4.2

%H R. Chapman, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00367-7">Moments of Dyck paths</a>, Discrete Math., 204 (1999), 113-117.

%H Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.

%H F. Disanto, A. Frosini, R. Pinzani and S. Rinaldi, <a href="http://arxiv.org/abs/math/0702550">A closed formula for the number of convex permutominoes</a>, arXiv:math/0702550 [math.CO], 2007.

%H Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">J. Int. Seq. 17 (2014) #14.1.5</a>.

%H Nikita Gogin and Mika Hirvensalo, <a href="https://pca-pdmi.ru/2020/files/10/GoHi2020ExtAbstract.pdf">On the Moments of Squared Binomial Coefficients</a>, (2020).

%H P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p45">Congruences of Finite Summations of the Coefficients in certain Generating Functions</a>, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.

%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

%H C. Jordan, <a href="/A002457/a002457_1.pdf">Calculus of Finite Differences</a>, Budapest, 1939. [Annotated scans of pages 448-450 only]

%H Bahar Kuloğlu, Engin Özkan, and Marin Marin, <a href="https://doi.org/10.2478/auom-2023-0023">Fibonacci and Lucas Polynomials in n-gon</a>, An. Şt. Univ. Ovidius Constanţa (Romania 2023) Vol. 31, No 2, 127-140.

%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)

%H A. Petojevic and N. Dapic, <a href="http://www.mi.sanu.ac.rs/~gvm/radovi/AP-Budva.pdf">The vAm(a,b,c;z) function</a>, Preprint 2013.

%H H. E. Salzer, <a href="http://dx.doi.org/10.1002/sapm1943221115">Coefficients for numerical differentiation with central differences</a>, J. Math. Phys., 22 (1943), 115-135.

%H H. E. Salzer, <a href="/A002457/a002457_2.pdf">Coefficients for numerical differentiation with central differences</a>, J. Math. Phys., 22 (1943), 115-135. [Annotated scanned copy]

%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].

%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)

%H L. W. Shapiro, W.-J. Woan and S. Getu, <a href="http://dx.doi.org/10.1137/0604046">Runs, slides and moments</a>, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.

%H Andrei K. Svinin, <a href="https://arxiv.org/abs/1610.05387">On some class of sums</a>, arXiv:1610.05387 [math.CO], 2016. See p. 5.

%H T. R. Van Oppolzer, <a href="http://www.archive.org/stream/lehrbuchzurbahnb02oppo#page/21/mode/1up">Lehrbuch zur Bahnbestimmung der Kometen und Planeten</a>, Vol. 2, Engelmann, Leipzig, 1880, p. 21.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CentralBetaFunction.html">Central Beta Function</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiFormulas.html">Pi Formulas</a>

%H Y. Q. Zhao, <a href="http://mathstat.carleton.ca/~zhao/TEACHING/70.265/random-v/random-v.html">Introduction to Probability with Applications</a>

%F G.f.: (1-4x)^(-3/2) = 1F0(3/2;;4x).

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

%F a(n-1) = 4^(n-1)*Sum_{i=0..n-1} binomial(n-1+i, i)*(n-i)/2^(n-1+i).

%F a(n) ~ 2*Pi^(-1/2)*n^(1/2)*2^(2*n)*{1 + 3/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 21 2001

%F (2*n+2)!/(2*n!*(n+1)!) = (n+n+1)!/(n!*n!) = 1/beta(n+1, n+1) in A061928.

%F Sum_{i=0..n} i * binomial(n, i)^2 = n*binomial(2*n, n)/2. - Yong Kong (ykong(AT)curagen.com), Dec 26 2000

%F a(n) ~ 2*Pi^(-1/2)*n^(1/2)*2^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 07 2002

%F a(n) = 1/Integral_{x=0..1} x^n (1-x)^n dx. - Fred W. Helenius (fredh(AT)ix.netcom.com), Jun 10 2003

%F E.g.f.: exp(2*x)*((1+4*x)*BesselI(0, 2*x) + 4*x*BesselI(1, 2*x)). - _Vladeta Jovovic_, Sep 22 2003

%F a(n) = Sum_{i+j+k=n} binomial(2i, i)*binomial(2j, j)*binomial(2k, k). - _Benoit Cloitre_, Nov 09 2003

%F a(n) = (2*n+1)*A000984(n) = A005408(n)*A000984(n). - _Zerinvary Lajos_, Dec 12 2010

%F a(n-1) = Sum_{k=0..n} A039599(n,k)*A000217(k), for n >= 1. - _Philippe Deléham_, Jun 10 2007

%F Sum of (n+1)-th row terms of triangle A132818. - _Gary W. Adamson_, Sep 02 2007

%F Sum_{n>=0} 1/a(n) = 2*Pi/3^(3/2). - _Jaume Oliver Lafont_, Mar 07 2009

%F a(n) = Sum_{k=0..n} binomial(2k,k)*4^(n-k). - _Paul Barry_, Apr 26 2009

%F a(n) = A000217(n) * A000108(n). - _David Scambler_, Nov 25 2010

%F a(n) = f(n, n-3) where f is given in A034261.

%F a(n) = A005430(n+1)/2 = A002011(n)/4.

%F a(n) = binomial(2n+2, 2) * binomial(2n, n) / binomial(n+1, 1), a(n) = binomial(n+1, 1) * binomial(2n+2, n+1) / binomial(2, 1) = binomial(2n+2, n+1) * (n+1)/2. - _Rui Duarte_, Oct 08 2011

%F G.f.: (G(0) - 1)/(4*x) where G(k) = 1 + 2*x*((2*k + 3)*G(k+1) - 1)/(k + 1). - _Sergei N. Gladkovskii_, Dec 03 2011 [Edited by _Michael Somos_, Dec 06 2013]

%F G.f.: 1 - 6*x/(G(0)+6*x) where G(k) = 1 + (4*x+1)*k - 6*x - (k+1)*(4*k-2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Aug 13 2012

%F G.f.: Q(0), where Q(k) = 1 + 4*(2*k + 1)*x*(2*k + 2 + Q(k+1))/(k+1). - _Sergei N. Gladkovskii_, May 10 2013 [Edited by _Michael Somos_, Dec 06 2013]

%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 4*x*(2*k+3)/(4*x*(2*k+3) + 2*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 06 2013

%F a(n) = 2^(4n)/Sum_{k=0..n} (-1)^k*C(2n+1,n-k)/(2k+1). - _Mircea Merca_, Nov 12 2013

%F a(n) = (2*n)!*[x^(2*n)] HeunC(0,0,-2,-1/4,7/4,4*x^2) where [x^n] f(x) is the coefficient of x^n in f(x) and HeunC is the Heun confluent function. - _Peter Luschny_, Nov 22 2013

%F 0 = a(n) * (16*a(n+1) - 2*a(n+2)) + a(n+1) * (a(n+2) - 6*a(n+1)) for all n in Z. - _Michael Somos_, Dec 06 2013

%F a(n) = 4^n*binomial(n+1/2, 1/2). - _Peter Luschny_, Apr 24 2014

%F a(n) = 4^n*hypergeom([-2*n,-2*n-1,1/2],[-2*n-2,1],2)*(n+1)*(2*n+1). - _Peter Luschny_, Sep 22 2014

%F a(n) = 4^n*hypergeom([-n,-1/2],[1],1). - _Peter Luschny_, May 19 2015

%F a(n) = 2*4^n*Gamma(3/2+n)/(sqrt(Pi)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015

%F Sum_{n >= 0} 2^(n+1)/a(n) = Pi, related to Newton/Euler's Pi convergence transformation series. - _Tony Foster III_, Jul 28 2016. See the Weisstein Pi link, eq. (23). - _Wolfdieter Lang_, Aug 26 2016

%F Boas-Buck recurrence: a(n) = (6/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, and a(0) = 1. Proof from a(n) = A046521(n+1,1). See comment in A046521. - _Wolfdieter Lang_, Aug 10 2017

%F a(n) = (1/3)*Sum_{i = 0..n+1} C(n+1,i)*C(n+1,2*n+1-i)*C(3*n+2-i,n+1) = (1/3)*Sum_{i = 0..2*n+1} (-1)^(i+1)*C(2*n+1,i)*C(n+i+1,i)^2. - _Peter Bala_, Feb 07 2018

%F a(n) = (2*n+1)*binomial(2*n, n). - _Kolosov Petro_, Apr 16 2018

%F a(n) = (-4)^n*binomial(-3/2, n). - _Peter Luschny_, Oct 23 2018

%F a(n) = 1 / Sum_{s=0..n} (-1)^s * binomial(n, s) / (n+s+1). - _Kolosov Petro_, Jan 22 2019

%F a(n) = Sum_{k = 0..n} (2*k + 1)*binomial(2*n + 1, n - k). - _Peter Bala_, Feb 25 2019

%F 4^n/a(n) = Integral_{x=0..1} (1 - x^2)^n. - _Michael Somos_, Jun 13 2019

%F D-finite with recurrence: 0 = a(n)*(6 + 4*n) - a(n+1)*(n + 1) for all n in Z. - _Michael Somos_, Jun 13 2019

%F Sum_{n>=0} (-1)^n/a(n) = 4*arcsinh(1/2)/sqrt(5). - _Amiram Eldar_, Sep 10 2020

%F From _Jianing Song_, Apr 10 2022: (Start)

%F G.f. for {1/a(n)}: 4*arcsin(sqrt(x)/2) / sqrt(x*(4-x)).

%F E.g.f. for {1/a(n)}: exp(x/4)*sqrt(Pi/x)*erf(sqrt(x)/2). (End)

%F G.f. for {1/a(n)}: 4*arctan(sqrt(x/(4-x))) / sqrt(x*(4-x)). - _Michael Somos_, Jun 17 2023

%F a(n) = Sum_{k = 0..n} (-1)^(n+k) * (n + 2*k + 1)*binomial(n+k, k). This is the particular case m = 1 of the identity Sum_{k = 0..m*n} (-1)^k * (n + 2*k + 1) * binomial(n+k, k) = (-1)^(m*n) * (m*n + 1) * binomial((m+1)*n+1, n). Cf. A090816 and A306290. - _Peter Bala_, Nov 02 2024

%F a(n) = (1/Pi)*(2*n + 1)*(2^(2*n + 1))*Integral_{x=0..oo} 1/(x^2 + 1)^(n + 1) dx. - _Velin Yanev_, Jan 28 2025

%e G.f. = 1 + 6*x + 30*x^2 + 140*x^3 + 630*x^4 + 2772*x^5 + 12012*x^6 + 51480*x^7 + ...

%p A002457:=n->(n+1) * binomial(2*(n+1),(n+1)) / 2; seq(A002457(n), n=0..50);

%p seq((2*n)!*coeff(series(HeunC(0,0,-2,-1/4,7/4,4*x^2),x,2*n+1),x,2*n),n=0..22); # _Peter Luschny_, Nov 22 2013

%t a[n_]:=(2*n+1)!/n!^2; Array[f, 23, 0] (* _Vladimir Joseph Stephan Orlovsky_, Dec 13 2008 *)

%o (PARI) {a(n) = if( n<0, 0, (2*n + 1)! / n!^2)}; /* _Michael Somos_, Dec 09 2002 */

%o (PARI) a(n) = (2*n+1)*binomial(2*n, n); \\ _Altug Alkan_, Apr 16 2018

%o (Haskell)

%o a002457 n = a116666 (2 * n + 1) (n + 1)

%o -- _Reinhard Zumkeller_, Nov 02 2013

%o (Sage)

%o A002457 = lambda n: binomial(n+1/2,1/2)<<2*n

%o [A002457(n) for n in range(23)] # _Peter Luschny_, Sep 22 2014

%o (Magma) [Factorial(2*n+1)/Factorial(n)^2: n in [0..25]]; // _Vincenzo Librandi_, Oct 12 2015

%Y Cf. A000531 (Banach's original match problem).

%Y Cf. A033876, A000984, A001803, A132818, A046521 (second column).

%Y Cf. A000108, A000217.

%Y A diagonal of A331430.

%Y The rightmost diagonal of the triangle A331431.

%K nonn,easy,nice,changed

%O 0,2

%A _N. J. A. Sloane_